Fractional nonlinear Schrödinger equation

Abstract

We consider the Cauchy problem for the fractional nonlinear Schrödinger equation

$$\begin{aligned} \left\{ \begin{array}{ll} i\partial _{t}u+\frac{2}{3}\left| \partial _{x}\right| ^{\frac{3}{2} }u=\lambda \left| u\right| ^{2}u,\,\, t>0, &{}\quad x\in \mathbb {R},\\ u\left( 1,x\right) =u_{0}\left( x\right) ,&{}\quad x\in \mathbb {R}. \end{array}\right. \end{aligned}$$

We develop the factorization technique to obtain the large-time asymptotic behavior of solutions which has a logarithmic phase modifications for large time comparing with the linear problem.

1 Introduction

We study the Cauchy problem for the fractional nonlinear Schrödinger equation

$$\begin{aligned} \left\{ \begin{array}{ll} i\partial _{t}u+\frac{2}{3}\left| \partial _{x}\right| ^{\frac{3}{2} }u=\lambda \left| u\right| ^{2}u,\,\, t>0, &{}\quad x\in \mathbb {R},\\ u\left( 0,x\right) =u_{0}\left( x\right) , &{}\quad x\in \mathbb {R}, \end{array} \right. \end{aligned}$$

(1.1)

where $\lambda \in \mathbb {R}$, the fractional derivative $\left| \partial _{x}\right| ^{\alpha }=\mathcal {F}^{-1}\left| \xi \right| ^{\alpha }\mathcal {F}$, here and below $\mathcal {F}$ stands for the Fourier transform $\hat{\phi }(\xi )=\frac{1}{\sqrt{2\pi }}\int _{\mathbb {R}}e^{-ix\xi }\phi (x)\mathrm{d}x$, and $\mathcal {F}^{-1}$ is the inverse Fourier transformation $\mathcal {F}^{-1}\phi =\frac{1}{\sqrt{2\pi }}\int _{\mathbb {R}}e^{ix\xi }\phi (\xi )d\xi$. Fractional nonlinear Schrödinger equation (1.1) appeared in [25, 26] with applications in quantum mechanics. Later, it was derived in various areas such as plasma physics, optimization, finance, free boundary obstacle problems, population dynamics and minimal surfaces. The case of fractional derivative $\left| \partial _{x}\right| ^{\frac{3}{2}}$ has a particular relevance to the two-dimensional water waves with surface tension (see [19, 20]). Recently fractional nonlinear Schrödinger equations attracted much attention of many authors, (see [3, 5, 6, 9,10,11, 13, 21, 22, 24] and references cited therein).

For the fractional nonlinear Schrödinger equations

$$\begin{aligned} i\partial _{t}u+\frac{1}{\alpha }\left| \partial _{x}\right| ^{\alpha }u=\lambda \left| u\right| ^{2}u, \end{aligned}$$

(1.2)

the local well posedness in $\mathbf {H}^{s}$ for $s\ge \frac{2-\alpha }{4}$ and ill posedness in $\mathbf {H}^{s}$ for $\frac{2-3\alpha }{4\left( \alpha +1\right) }<s<\frac{2-\alpha }{4}$, $1<\alpha <2$, were obtained in [6] through the multilinear estimates based on the Bourgain spaces. Hence, the result in [6] shows that global $\mathbf {L}^{2}$ well posedness fails for the cubic nonlinearity, when $1<\alpha <2$. In [17], the local well posedness and ill-posedness were also considered for (1.2) with $0<\alpha <2$, $\alpha \ne 1$ and general nonlinearity $\lambda \left| u\right| ^{p-1}u$ in the scaling invariant Sobolev spaces $\mathbf {H}^{\frac{1}{2}-\frac{\alpha }{p-1}}$. In particular, the small data scattering in $\mathbf {H}^{\frac{1}{2}-\frac{\alpha }{p-1}}$ was shown for the case of $p\ge 5$. Cubic nonlinearities often require some logarithmic phase corrections in the large-time asymptotics comparing to the corresponding linear problem. Our purpose in the present paper is to show that the factorization technique originated in papers [14,15,16, 27, 28] can also be developed for the fractional nonlinear Schrödinger equation (1.1).

We introduce Notation and Function Spaces. $\mathbf {L}^{p}=\left\{ \phi \in \mathbf {S}^{\prime };\left\| \phi \right\| _{\mathbf {L}^{p}} <\infty \right\}$ is the usual Lebesgue space, and the norm is defined by $\left\| \phi \right\| _{\mathbf {L}^{p}}=\left( \int _{\mathbb {R} }\left| \phi \left( x\right) \right| ^{p}\mathrm{d}x\right) ^{\frac{1}{p}}$ for $1\le p<\infty$ and $\left\| \phi \right\| _{\mathbf {L}^{\infty } }=\sup _{x\in \mathbb {R}}\left| \phi \left( x\right) \right|$ for $p=\infty$. The weighted Sobolev space is

$$\begin{aligned} \mathbf {H}_{p}^{m,s}=\left\{ \varphi \in \mathbf {S}^{\prime };\left\| \phi \right\| _{\mathbf {H}_{p}^{m,s}}=\left\| \left\langle x\right\rangle ^{s}\left\langle i\partial _{x}\right\rangle ^{m}\phi \right\| _{\mathbf {L} ^{p}}<\infty \right\} , \end{aligned}$$

$m,s\in \mathbb {R},1\le p\le \infty$, $\left\langle x\right\rangle =\sqrt{1+x^{2}},\left\langle i\partial _{x}\right\rangle =\sqrt{1-\partial _{x}^{2}}$. We also use the notations $\mathbf {H}^{m,s}=\mathbf {H}_{2} ^{m,s}$, $\mathbf {H}^{m}=\mathbf {H}^{m,0}$ shortly, if it does not cause any confusion. Let $\mathbf {C}(\mathbf {I};\mathbf {B})$ be the space of continuous functions from an interval $\mathbf {I}$ to a Banach space $\mathbf {B}$.

To state our main result, we introduce the dilation operator $\mathcal {D} _{t}\phi \left( x\right) =t^{-\frac{1}{2}}\phi \left( \frac{x}{t}\right)$, the scaling operator $\left( \mathcal {B}\phi \right) \left( x\right) =\phi \left( \mu \left( x\right) \right)$ and the multiplication factor $M\left( t,\eta \right) =e^{\frac{it}{3}\left| \eta \right| ^{\frac{3}{2}}}$, where the stationary point $\mu \left( x\right) =x\left| x\right|$. We are in a position to state the main result of this paper.

Theorem 1.1

Let the initial data $u_{0}\in \mathbf {H}^{2}\cap \mathbf {H}^{0,1}$ have a small norm $\left\| u_{0}\right\| _{\mathbf {H}^{2}\cap \mathbf {H}^{0,1}}$. Then, there exists a unique global solution u of the Cauchy problem (1.1) such that $u\in \mathbf {C} \left( \left[ 0,\infty \right) ;\mathbf {H}^{2}\cap \mathbf {H}^{0,1}\right)$. Also the time decay estimate $\left\| u\left( t\right) \right\| _{\mathbf {L}^{\infty }}\le C\left( 1+t\right) ^{-\frac{1}{2}}$ is true. Moreover, there exists a unique modified final state $W_{+}\in \mathbf {L} ^{\infty }$ such that the asymptotics

$$\begin{aligned} u\left( t\right) =\mathcal {D}_{t}\mathcal {B}MW_{+}\exp \left( -i\lambda \left| W_{+}\right| ^{2}\log t\right) +O\left( \varepsilon t^{-\delta }\right) \end{aligned}$$

(1.3)

is valid for $t\rightarrow \infty$ uniformly with respect to $x\in \mathbb {R}$, where $\delta >0$.

For the convenience of the reader, we now give a sketch of the proof. First, by using the factorization techniques, we change $u=\mathcal {D}_{t}\mathcal {B} M\mathcal {Q}\widehat{\varphi }$ so that Eq. (1.1) takes the form $i\partial _{t}\widehat{\varphi }=\lambda t^{-1}\mathcal {Q}^{*}\left( \left| \mathcal {Q}\widehat{\varphi }\right| ^{2}\mathcal {Q} \widehat{\varphi }\right)$, where the direct defect operator $\mathcal {Q} \left( t\right) \phi =\sqrt{\frac{t}{2\pi }}\int _{\mathbb {R}}e^{-itS\left( \xi ,\eta \right) }\phi \left( \xi \right) d\xi$, the conjugate defect operator $\mathcal {Q}^{*}\left( t\right) \phi =\sqrt{\frac{t}{2\pi }}\int _{\mathbb {R}}e^{itS\left( \xi ,\eta \right) }\phi \left( \eta \right) \Lambda ^{\prime \prime }\left( \eta \right) d\eta$, the phase function $S\left( \xi ,\eta \right) =\Lambda \left( \xi \right) -\Lambda \left( \eta \right) -\Lambda ^{\prime }\left( \eta \right) \left( \xi -\eta \right)$ and the symbol $\Lambda \left( \xi \right) =\frac{2}{3}\left| \xi \right| ^{\frac{3}{2}}$. Then, the most difficulty is to estimate derivatives of the defect operators $\mathcal {Q}$ and $\mathcal {Q}^{*}$. For this purpose, we apply the $\mathbf {L}^{2}$-theory of pseudodifferential operators.

The rest of the paper is organized as follows. In Sect. 2, we prove some preliminary estimates for the defect operators in the uniform metrics and $\mathbf {L}^{2}$-norm. Section 3 is devoted to the proof of the a priori estimates for the local solutions. We prove Theorem 1.1 in Sect. 4.

2 Preliminaries

2.1 Factorization techniques

Denote the symbol $\Lambda \left( \xi \right) =\frac{2}{3}\left| \xi \right| ^{\frac{3}{2}}$, then the free evolution group has the form $\mathcal {U}\left( t\right) =\mathcal {F}^{-1}e^{-it\Lambda \left( \xi \right) }\mathcal {F}$. We have $\mathcal {U}\left( t\right) \mathcal {F}^{-1}\phi =\frac{1}{\sqrt{2\pi }}\int _{\mathbb {R}}e^{it\left( \frac{x}{t}\xi -\Lambda \left( \xi \right) \right) }\phi \left( \xi \right) d\xi$. Consider the stationary point $\mu \left( x\right)$ defined by the equation $\Lambda ^{\prime }\left( \mu \right) =x$. Since $\Lambda ^{\prime \prime }\left( \xi \right) =\frac{1}{2}\left| \xi \right| ^{-\frac{1}{2}}>0$, then $\Lambda ^{\prime }\left( \xi \right) =\left| \xi \right| ^{-\frac{1}{2}}\xi$ is monotonous. Hence, there exists a unique stationary point $\mu \left( x\right) =x\left| x\right|$ such that $\Lambda ^{\prime }\left( \mu \left( x\right) \right) =x$ for all $x\in \mathbb {R}$. Then, we write

$$\begin{aligned} \mathcal {U}\left( t\right) \mathcal {F}^{-1}\phi&= \mathcal {D}_{t} \sqrt{\frac{t}{2\pi }}e^{\frac{it}{3}\left| \eta \right| ^{\frac{3}{2}} }\int _{\mathbb {R}}e^{-it\left( \Lambda \left( \xi \right) -\Lambda \left( \mu \left( x\right) \right) -x\left( \xi -\mu \left( x\right) \right) \right) }\phi \left( \xi \right) \mathrm{d}\xi \\&=\mathcal {D}_{t}\mathcal {B}M\sqrt{\frac{t}{2\pi }}\int _{\mathbb {R} }e^{-itS\left( \xi ,\eta \right) }\phi \left( \xi \right) \mathrm{d}\xi =\mathcal {D} _{t}\mathcal {B}M\mathcal {Q}\phi , \end{aligned}$$

where the dilation operator $\mathcal {D}_{t}\phi \left( x\right) =t^{-\frac{1}{2}}\phi \left( \frac{x}{t}\right)$, the scaling operator $\left( \mathcal {B}\phi \right) \left( x\right) =\phi \left( \mu \left( x\right) \right)$, the multiplication factor $M\left( t,\eta \right) =e^{\frac{it}{3}\left| \eta \right| ^{\frac{3}{2}}}$, the phase function $S\left( \xi ,\eta \right) =\frac{2}{3}\left| \xi \right| ^{\frac{3}{2}}-\frac{2}{3}\left| \eta \right| ^{\frac{3}{2}}-\left| \eta \right| ^{-\frac{1}{2}}\eta \left( \xi -\eta \right)$ and the defect operator $\mathcal {Q}\left( t\right) \phi =\sqrt{\frac{t}{2\pi }} \int _{\mathbb {R}}e^{-itS\left( \xi ,\eta \right) }\phi \left( \xi \right) \mathrm{d}\xi$. Also, we define the conjugate defect operator $\mathcal {Q}^{*}\left( t\right) \phi =\sqrt{\frac{t}{2\pi }}\int _{\mathbb {R}}e^{itS\left( \xi ,\eta \right) }\phi \left( \eta \right) \Lambda ^{\prime \prime }\left( \eta \right) \mathrm{d}\eta$. Thus, we have the representation for the free evolution group $\mathcal {U}\left( t\right) \mathcal {F}^{-1}=\mathcal {D} _{t}\mathcal {B}M\mathcal {Q}$ and for the inverse evolution group $\mathcal {FU}\left( -t\right) =\mathcal {Q}^{*}\overline{M}\mathcal {B} ^{-1}\mathcal {D}_{t}^{-1}$ with the inverse scaling operator $\left( \mathcal {B}^{-1}\phi \right) \left( \eta \right) =\phi \left( \Lambda ^{\prime }\left( \eta \right) \right)$ and the inverse dilation operator $\mathcal {D}_{t}^{-1}\phi \left( x\right) =t^{\frac{1}{2}}\phi \left( xt\right)$. We define the new dependent variable $\widehat{\varphi }=$ $\mathcal {FU}\left( -t\right) u\left( t\right)$. Since $\mathcal {FU} \left( -t\right) \mathcal {L}=i\partial _{t}\mathcal {FU}\left( -t\right)$, $\mathcal {L}=i\partial _{t}+\frac{2}{3}\left| \partial _{x}\right| ^{\frac{3}{2}}$, applying the operator $\mathcal {FU}\left( -t\right)$ to Eq. (1.1) and substituting $u\left( t\right) =\mathcal {U}\left( t\right) \mathcal {F}^{-1}\widehat{\varphi }=\mathcal {D}_{t}\mathcal {B} M\mathcal {Q}\widehat{\varphi }$, we find the following factorization property

$$\begin{aligned} i\partial _{t}\widehat{\varphi }&=i\partial _{t}\mathcal {FU}\left( -t\right) u=\mathcal {FU}\left( -t\right) \mathcal {L}u=\mathcal {FU}\left( -t\right) \left( \lambda \left| u\right| ^{2}u\right) \\&=t^{-1}\mathcal {Q}^{*}\overline{M}\mathcal {B}^{-1}\left( \lambda \left| \mathcal {B}M\mathcal {Q}\widehat{\varphi }\right| ^{2} \mathcal {B}M\mathcal {Q}\widehat{\varphi }\right) =\lambda t^{-1} \mathcal {Q}^{*}\left| v\right| ^{2}v, \end{aligned}$$

where $v=\mathcal {Q}\widehat{\varphi }$.

2.2 Estimates for defect operator $\mathcal {Q}$ in the uniform metrics

Define the kernel $A\left( t,\eta \right) =\sqrt{\frac{t}{2\pi }} \int _{\mathbb {R}}e^{-itS\left( \xi ,\eta \right) }\left\langle \xi \right\rangle ^{-1}\mathrm{d}\xi$, where $S\left( \xi ,\eta \right) =\frac{2}{3}\left| \xi \right| ^{\frac{3}{2}}-\frac{2}{3}\left| \eta \right| ^{\frac{3}{2}}-\left| \eta \right| ^{-\frac{1}{2}} \eta \left( \xi -\eta \right)$. We change $\xi =\eta y$, then

$$\begin{aligned} A\left( t,\eta \right) =\left| \eta \right| \sqrt{\frac{t}{2\pi }} \int _{\mathbb {R}}e^{-it\left| \eta \right| ^{\frac{3}{2}}\left( \frac{2}{3}\left| y\right| ^{\frac{3}{2}}+\frac{1}{3}-y\right) }\left\langle \eta y\right\rangle ^{-1}\mathrm{d}y. \end{aligned}$$

To compute the asymptotics of the kernel $A\left( t,\eta \right)$ for large time, we apply the stationary phase method (see [12], p. 110)

$$\begin{aligned} \int _{\mathbb {R}}e^{itg\left( y\right) }f\left( y\right) \mathrm{d}y=e^{izg\left( y_{0}\right) }f\left( y_{0}\right) \sqrt{\frac{2\pi }{t\left| g^{\prime \prime }\left( y_{0}\right) \right| }}e^{i\frac{\pi }{4}\text {sgn}g^{\prime \prime }\left( y_{0}\right) }+O\left( t^{-\frac{3}{2} }\right) \end{aligned}$$

(2.1)

for $t\rightarrow \infty$, where the stationary point $y_{0}$ is defined by the equation $g^{\prime }\left( y_{0}\right) =0$. By virtue of formula (2.1) with $g\left( y\right) =-\left( \frac{2}{3}\left| y\right| ^{\frac{3}{2}}+\frac{1}{3}-y\right)$, $f\left( y\right) =\left\langle \eta y\right\rangle ^{-1}$, $y_{0}=1$, we get $A\left( t,\eta \right) =\frac{\left\langle \eta \right\rangle ^{-1}}{\sqrt{i\Lambda ^{\prime \prime }\left( \eta \right) }}+O\left( t^{\frac{1}{2} }\left| \eta \right| \left\langle t\left| \eta \right| ^{\frac{3}{2}}\right\rangle ^{-\frac{3}{2}}\right)$ for $t\left| \eta \right| ^{\frac{3}{2}}\rightarrow \infty$. In the next lemma, we find the large-time asymptotics for the defect operator $\mathcal {Q}\phi$. Denote $\left\{ \eta \right\} =\frac{\left| \eta \right| }{\left\langle \eta \right\rangle }$.

Lemma 2.1

The estimate

$$\begin{aligned} \left\| \mathcal {Q}\phi -A\left( t,\eta \right) \left\langle \eta \right\rangle \phi \left( \eta \right) \right\| _{\mathbf {L}^{\infty }}\le Ct^{-\frac{1}{4}}\left( t^{\gamma -\frac{1}{4}}+\left\{ \eta \right\} ^{\frac{3}{8}}\right) \left( \left\| \left\langle \xi \right\rangle \phi _{\xi }\right\| _{\mathbf {L}^{2}}+\left\| \phi \right\| _{\mathbf {L}^{2}}\right) \end{aligned}$$

is valid for all $t\ge 1$, where $\gamma >0$ is small.

Proof

We integrate by parts via the identity $e^{-itS\left( \xi ,\eta \right) }=H_{1}\partial _{\xi }\left( \left( \xi -\eta \right) e^{-itS\left( \xi ,\eta \right) }\right)$ with $H_{1}=\left( 1-it\left( \xi -\eta \right) \partial _{\xi }S\left( \xi ,\eta \right) \right) ^{-1}$

$$\begin{aligned} \mathcal {Q}\phi -A\left( t,\eta \right) \left\langle \eta \right\rangle \phi \left( \eta \right)&=-\sqrt{\frac{t}{2\pi }}\int _{\mathbb {R}}e^{-itS\left( \xi ,\eta \right) }\frac{\left\langle \xi \right\rangle \phi \left( \xi \right) -\left\langle \eta \right\rangle \phi \left( \eta \right) }{\xi -\eta }\left( \xi -\eta \right) ^{2}\partial _{\xi }\left( H_{1}\left\langle \xi \right\rangle ^{-1}\right) \mathrm{d}\xi \\&\quad -\sqrt{\frac{t}{2\pi }}\int _{\mathbb {R}}e^{-itS\left( \xi ,\eta \right) }\left( \xi -\eta \right) H_{1}\left\langle \xi \right\rangle ^{-1} \partial _{\xi }\left( \left\langle \xi \right\rangle \phi \left( \xi \right) \right) \mathrm{d}\xi . \end{aligned}$$

Since $\partial _{\xi }S\left( \xi ,\eta \right) =\left| \xi \right| ^{-\frac{1}{2}}\xi -\left| \eta \right| ^{-\frac{1}{2}}\eta$, we get the estimates

$$\begin{aligned} \left| \left( \xi -\eta \right) H_{1}\left\langle \xi \right\rangle ^{-1}\right| +\left| \left( \xi -\eta \right) ^{2}\partial _{\xi }\left( H_{1}\left\langle \xi \right\rangle ^{-1}\right) \right| \le \frac{C\left| \xi -\eta \right| \left\langle \xi \right\rangle ^{-1} }{1+t\left| \xi -\eta \right| \left| \left| \xi \right| ^{-\frac{1}{2}}\xi -\left| \eta \right| ^{-\frac{1}{2}}\eta \right| }. \end{aligned}$$

Hence, by the Hardy inequality, we obtain

$$\begin{aligned} \left| \mathcal {Q}\phi -A\left( t,\eta \right) \left\langle \eta \right\rangle \phi \left( \eta \right) \right| \le Ct^{\frac{1}{2} }\left( \left\| \left\langle \xi \right\rangle \phi _{\xi }\right\| _{\mathbf {L}^{2}}+\left\| \phi \right\| _{\mathbf {L}^{2}}\right) I^{\frac{1}{2}}, \end{aligned}$$

where

$$\begin{aligned} I=\int _{\mathbb {R}}\frac{\left( \xi -\eta \right) ^{2}\left\langle \xi \right\rangle ^{-2}\mathrm{d}\xi }{\left( 1+t\left| \xi -\eta \right| \left| \left| \xi \right| ^{-\frac{1}{2}}\xi -\left| \eta \right| ^{-\frac{1}{2}}\eta \right| \right) ^{2}}. \end{aligned}$$

We have $\left| \left| \xi \right| ^{-\frac{1}{2}}\xi -\left| \eta \right| ^{-\frac{1}{2}}\eta \right| =\frac{\left| \xi -\eta \right| }{\left| \xi \right| ^{\frac{1}{2}}+\left| \eta \right| ^{\frac{1}{2}}}\theta \left( \xi \eta \right) +\left( \left| \xi \right| ^{\frac{1}{2}}+\left| \eta \right| ^{\frac{1}{2}}\right) \theta \left( -\xi \eta \right)$, where $\theta \left( x\right)$ is the Heaviside function. Hence, we find $I=I_{1}+I_{2}$, where

$$\begin{aligned} I_{1}=\int _{\mathbb {R}}\frac{\left( \xi -\eta \right) ^{2}\left\langle \xi \right\rangle ^{-2}\theta \left( \xi \eta \right) \mathrm{d}\xi }{\left( 1+t\frac{\left( \xi -\eta \right) ^{2}}{\left| \xi \right| ^{\frac{1}{2}}+\left| \eta \right| ^{\frac{1}{2}}}\right) ^{2}},I_{2} =\int _{\mathbb {R}}\frac{\left( \xi -\eta \right) ^{2}\left\langle \xi \right\rangle ^{-2}\theta \left( -\xi \eta \right) \mathrm{d}\xi }{\left( 1+t\left| \xi -\eta \right| \left| \left| \xi \right| ^{\frac{1}{2}}+\left| \eta \right| ^{\frac{1}{2}}\right| \right) ^{2}}. \end{aligned}$$

For $\eta >0$, we have

$$\begin{aligned} I_{1}&\le \frac{C\eta ^{2}}{1+t^{2}\eta ^{3}}\int _{0}^{\frac{\eta }{2}} \mathrm{d}\xi +C\left\langle \eta \right\rangle ^{-2}\int _{\frac{\eta }{2}}^{2\eta } \frac{\left( \xi -\eta \right) ^{2}\mathrm{d}\xi }{\left( 1+t\eta ^{-\frac{1}{2}}\left( \xi -\eta \right) ^{2}\right) ^{2}} \\&\quad +Ct^{2\gamma -2}\int _{2\eta }^{\infty }\frac{\mathrm{d}\xi }{\xi ^{1-3\gamma }\left\langle \xi \right\rangle ^{2}} \le Ct^{2\gamma -2}+\frac{C\eta ^{3}\left\langle \eta \right\rangle ^{-2}}{\left\langle t\eta ^{\frac{3}{2}}\right\rangle ^{\frac{3}{2}}}\le Ct^{2\gamma -2}+Ct^{-\frac{3}{2}}\left\{ \eta \right\} ^{\frac{3}{4}} \end{aligned}$$

and

$$\begin{aligned} I_{2}\le Ct^{2\gamma -2}\int _{0}^{1}\frac{\mathrm{d}\xi }{\left( \left| \xi \right| +\left| \eta \right| \right) ^{1-3\gamma }}+Ct^{-2} \int _{1}^{\infty }\xi ^{-3}\mathrm{d}\xi \le Ct^{2\gamma -2}. \end{aligned}$$

Thus, we get

$$\begin{aligned} \left| \mathcal {Q}\phi -A\left( t,\eta \right) \left\langle \eta \right\rangle \phi \left( \eta \right) \right| \le C\left( t^{\gamma -\frac{1}{2}}+t^{-\frac{1}{4}}\left\{ \eta \right\} ^{\frac{3}{8} }\right) \left( \left\| \left\langle \xi \right\rangle \phi _{\xi }\right\| _{\mathbf {L}^{2}}+\left\| \phi \right\| _{\mathbf {L}^{2} }\right) . \end{aligned}$$

Lemma 2.1 is proved. $\square$

2.3 Estimates for conjugate defect operator $\mathcal {Q}^{*}$ in the uniform metrics

Define $\chi _{1}\left( x\right) \in \mathbf {C}^{4}\left( \mathbb {R}\right)$ such that $\chi _{1}\left( x\right) =1$ for $\left| x\right| \le \frac{1}{3}$ and $\chi _{1}\left( x\right) =0$ for $\left| x\right| \ge \frac{2}{3}$, $\chi _{2}\left( x\right) =1-\chi _{1}\left( x\right)$. Denote the conjugate kernel

$$\begin{aligned} A^{*}\left( t,\xi \right) =\sqrt{\frac{t}{2\pi }}\int _{\mathbb {R} }e^{itS\left( \xi ,\eta \right) }\chi _{2}\left( \eta t\right) \chi _{2}\left( \frac{\eta }{\xi }\right) \Lambda ^{\prime \prime }\left( \eta \right) \mathrm{d}\eta . \end{aligned}$$

By virtue of formula (2.1) with $g\left( y\right) =S\left( \xi ,y\right)$, $f\left( y\right) =\chi _{2}\left( yt\right) \chi _{2}\left( \frac{y}{\xi }\right) \Lambda ^{\prime \prime }\left( y\right)$, $y_{0}=\xi$, we obtain the asymptotics $A^{*}\left( t,\xi \right) =\chi _{2}\left( \xi t\right) \sqrt{i\Lambda ^{\prime \prime }\left( \xi \right) }\left( 1+O\left( t^{-1}\right) \right)$ for $t\rightarrow \infty$.

In the next lemma, we estimate the conjugate defect operator $\mathcal {Q}^{*}$.

Lemma 2.2

The estimate

$$\begin{aligned} \left\| \left\langle \xi \right\rangle ^{-\frac{1}{8}}\left( \mathcal {Q}^{*}\phi -A^{*}\phi \right) \right\| _{\mathbf {L}^{\infty }}&\le Ct^{-\frac{1}{4}}\left\| \left| \eta \right| ^{-\frac{1}{4} }\partial _{\eta }\phi \right\| _{\mathbf {L}^{2}\left( \left| \eta \right| \ge \frac{1}{3}t^{-1}\right) }\\&\quad +Ct^{-\frac{1}{4}}\left\| \left| \eta \right| ^{-\frac{5}{4}} \phi \right\| _{\mathbf {L}^{2}\left( \left| \eta \right| \ge \frac{1}{3}t^{-1}\right) }+C\left\| \phi \right\| _{\mathbf {L}^{\infty }\left( \left| \eta \right| \le t^{-1}\right) } \end{aligned}$$

is true for all $t\ge 1$.

Proof

We write

$$\begin{aligned} \mathcal {Q}^{*}\phi -A^{*}\phi&=\sqrt{\frac{t}{2\pi }}\int _{\mathbb {R} }e^{itS\left( \xi ,\eta \right) }\phi \left( \eta \right) \chi _{1}\left( \eta t\right) \Lambda ^{\prime \prime }\left( \eta \right) \mathrm{d}\eta \\&\quad +\sqrt{\frac{t}{2\pi }}\int _{\mathbb {R}}e^{itS\left( \xi ,\eta \right) } \phi \left( \eta \right) \chi _{2}\left( \eta t\right) \chi _{1}\left( \frac{\eta }{\xi }\right) \Lambda ^{\prime \prime }\left( \eta \right) \mathrm{d}\eta \\&\quad +\sqrt{\frac{t}{2\pi }}\int _{\mathbb {R}}e^{itS\left( \xi ,\eta \right) }\left( \phi \left( \eta \right) -\phi \left( \xi \right) \right) \chi _{2}\left( \eta t\right) \chi _{2}\left( \frac{\eta }{\xi }\right) \Lambda ^{\prime \prime }\left( \eta \right) \mathrm{d}\eta =I_{1}+I_{2}+I_{3}. \end{aligned}$$

For the first integral, we have $\left| I_{1}\right| \le Ct^{\frac{1}{2}}\left\| \phi \right\| _{\mathbf {L}^{\infty }\left( \left| \eta \right| \le t^{-1}\right) }\int _{\left| \eta \right| \le t^{-1}}\left| \eta \right| ^{-\frac{1}{2}}\mathrm{d}\eta \le C\left\| \phi \right\| _{\mathbf {L}^{\infty }\left( \left| \eta \right| \le t^{-1}\right) }$. In the second integral, we integrate by parts via the identity $e^{itS\left( \xi ,\eta \right) }=H_{2}\partial _{\eta }\left( \eta e^{itS\left( \xi ,\eta \right) }\right)$ with $H_{2}=\left( 1+it\eta \partial _{\eta }S\left( \xi ,\eta \right) \right) ^{-1}$, $\partial _{\eta }S\left( \xi ,\eta \right) =\frac{1}{2}\left| \eta \right| ^{-\frac{1}{2}}\left( \eta -\xi \right)$, then we get

$$\begin{aligned} I_{2}&=-\sqrt{\frac{t}{2\pi }}\int _{\mathbb {R}}e^{itS\left( \xi ,\eta \right) }\phi \left( \eta \right) \eta \partial _{\eta }\left( H_{2} \chi _{2}\left( \eta t\right) \chi _{1}\left( \frac{\eta }{\xi }\right) \Lambda ^{\prime \prime }\left( \eta \right) \right) \mathrm{d}\eta \\&\quad -\sqrt{\frac{t}{2\pi }}\int _{\mathbb {R}}e^{itS\left( \xi ,\eta \right) }\eta H_{2}\chi _{2}\left( \eta t\right) \chi _{1}\left( \frac{\eta }{\xi }\right) \Lambda ^{\prime \prime }\left( \eta \right) \partial _{\eta }\phi \left( \eta \right) \mathrm{d}\eta . \end{aligned}$$

We have the estimate

$$\begin{aligned} \left| \eta H_{2}\chi _{2}\left( \eta t\right) \chi _{1}\left( \frac{\eta }{\xi }\right) \Lambda ^{\prime \prime }\left( \eta \right) \right| +\left| \eta ^{2}\partial _{\eta }\left( H_{2}\chi _{2}\left( \eta t\right) \chi _{1}\left( \frac{\eta }{\xi }\right) \Lambda ^{\prime \prime }\left( \eta \right) \right) \right| \le \frac{C\left| \eta \right| ^{\frac{1}{2}}}{1+t\left| \eta \right| ^{\frac{1}{2}}\left| \xi \right| } \end{aligned}$$

in the domain $\frac{1}{3}t^{-1}\le \left| \eta \right| \le \left| \xi \right|$. Hence, we obtain

$$\begin{aligned} \left| I_{2}\right|&\le Ct^{\frac{1}{2}}\int _{\frac{1}{3} t^{-1}\le \left| \eta \right| \le \left| \xi \right| } \frac{\left( \left| \phi \left( \eta \right) \right| \left| \eta \right| ^{-1}+\left| \partial _{\eta }\phi \left( \eta \right) \right| \right) \left| \eta \right| ^{\frac{1}{2}}\mathrm{d}\eta }{1+t\left| \eta \right| ^{\frac{1}{2}}\left| \xi \right| }\\&\le Ct^{\frac{1}{2}}\left( \left\| \left| \eta \right| ^{-\frac{5}{4}}\phi \right\| _{\mathbf {L}^{2}\left( \left| \eta \right| \ge \frac{1}{3t}\right) }+\left\| \left| \eta \right| ^{-\frac{1}{4}}\partial _{\eta }\phi \right\| _{\mathbf {L} ^{2}\left( \left| \eta \right| \ge \frac{1}{3t}\right) }\right) I_{4}^{\frac{1}{2}}\\&\le Ct^{-\frac{1}{3}}\left( \left\| \left| \eta \right| ^{-\frac{5}{4}}\phi \right\| _{\mathbf {L}^{2}\left( \left| \eta \right| \ge \frac{1}{3t}\right) }+\left\| \left| \eta \right| ^{-\frac{1}{4}}\partial _{\eta }\phi \right\| _{\mathbf {L} ^{2}\left( \left| \eta \right| \ge \frac{1}{3t}\right) }\right) , \end{aligned}$$

since for $I_{4}=\int _{\frac{1}{3}t^{-1}\le \left| \eta \right| \le \left| \xi \right| }\frac{\left| \eta \right| ^{\frac{3}{2} }\mathrm{d}\eta }{\left( 1+t\left| \eta \right| ^{\frac{1}{2}}\left| \xi \right| \right) ^{2}}$, we have the estimate

$$\begin{aligned} I_{4} \le C\left| \xi \right| ^{\frac{5}{2}}\int _{\left| z\right| \le 1}\frac{\left| z\right| ^{\frac{3}{2}}\mathrm{d}z}{1+t^{2}\left| \xi \right| ^{3}\left| z\right| }\le Ct^{-\frac{5}{3}}\int _{\left| z\right| \le 1}\left| z\right| ^{\frac{2}{3}}\mathrm{d}z\le Ct^{-\frac{5}{3}}. \end{aligned}$$

Finally, in $I_{3}$, we integrate by parts via the identity $e^{itS\left( \xi ,\eta \right) }=H_{3}\partial _{\eta }\left( \left( \eta -\xi \right) e^{itS\left( \xi ,\eta \right) }\right)$ with $H_{3}=\left( 1+it\left( \eta -\xi \right) \partial _{\eta }S\left( \xi ,\eta \right) \right) ^{-1}$, then we get

$$\begin{aligned} I_{3}&=-\sqrt{\frac{t}{2\pi }}\int _{\mathbb {R}}e^{itS\left( \xi ,\eta \right) }\frac{\phi \left( \eta \right) -\phi \left( \xi \right) }{\eta -\xi }\left( \eta -\xi \right) ^{2}\partial _{\eta }\left( H_{3}\chi _{2}\left( \eta t\right) \chi _{2}\left( \frac{\eta }{\xi }\right) \Lambda ^{\prime \prime }\left( \eta \right) \right) \mathrm{d}\eta \\&\quad -\sqrt{\frac{t}{2\pi }}\int _{\mathbb {R}}e^{itS\left( \xi ,\eta \right) }\left( \eta -\xi \right) H_{3}\chi _{2}\left( \eta t\right) \chi _{2}\left( \frac{\eta }{\xi }\right) \Lambda ^{\prime \prime }\left( \eta \right) \partial _{\eta }\phi \left( \eta \right) \mathrm{d}\eta . \end{aligned}$$

We find the estimates

$$\begin{aligned} \left| \left( \eta -\xi \right) H_{3}\chi _{2}\left( \eta t\right) \chi _{2}\left( \frac{\eta }{\xi }\right) \Lambda ^{\prime \prime }\left( \eta \right) \right| +\left| \left( \eta -\xi \right) ^{2} \partial _{\eta }\left( H_{3}\chi _{2}\left( \eta t\right) \chi _{2}\left( \frac{\eta }{\xi }\right) \Lambda ^{\prime \prime }\left( \eta \right) \right) \right| \le \frac{C\left| \eta -\xi \right| \left| \eta \right| ^{-\frac{1}{2}}}{1+t\left| \eta \right| ^{-\frac{1}{2}}\left( \eta -\xi \right) ^{2}} \end{aligned}$$

in the domain $\frac{1}{3}\max \left( t^{-1},\left| \xi \right| \right) \le \left| \eta \right|$. Then, we obtain

$$\begin{aligned} \left| I_{3}\right|&\le Ct^{\frac{1}{2}}\int _{\frac{1}{3} \max \left( t^{-1},\left| \xi \right| \right) \le \left| \eta \right| }\left( \left| \frac{\phi \left( \eta \right) -\phi \left( \xi \right) }{\eta -\xi }\right| +\left| \partial _{\eta }\phi \left( \eta \right) \right| \right) \frac{\left| \eta -\xi \right| \left| \eta \right| ^{-\frac{1}{2}}\mathrm{d}\eta }{1+t\left| \eta \right| ^{-\frac{1}{2}}\left( \eta -\xi \right) ^{2}}\\&\le Ct^{\frac{1}{2}}\left( \left\| \left| \eta \right| ^{-\frac{1}{4}}\partial _{\eta }\phi \right\| _{\mathbf {L}^{2}\left( \left| \eta \right| \ge \frac{1}{3t}\right) }+\left\| \left| \eta \right| ^{-\frac{5}{4}}\phi \right\| _{\mathbf {L}^{2}\left( \left| \eta \right| \ge \frac{1}{3t}\right) }\right) I_{5}^{\frac{1}{2}}\\&\le C\left\langle \xi \right\rangle ^{\frac{1}{8}}t^{-\frac{1}{4}}\left( \left\| \left| \eta \right| ^{-\frac{1}{4}}\partial _{\eta } \phi \right\| _{\mathbf {L}^{2}\left( \left| \eta \right| \ge \frac{1}{3t}\right) }+\left\| \left| \eta \right| ^{-\frac{5}{4} }\phi \right\| _{\mathbf {L}^{2}\left( \left| \eta \right| \ge \frac{1}{3t}\right) }\right) , \end{aligned}$$

since for $I_{5}=\int _{\frac{1}{3}\max \left( t^{-1},\left| \xi \right| \right) \le \left| \eta \right| }\frac{\left( \eta -\xi \right) ^{2}\left| \eta \right| ^{-\frac{1}{2}}\mathrm{d}\eta }{\left( 1+t\left| \eta \right| ^{-\frac{1}{2}}\left( \eta -\xi \right) ^{2}\right) ^{2}}$, we have

$$\begin{aligned} I_{5}&\le C\left| \xi \right| ^{\frac{5}{2}}\int _{\frac{1}{3} \le \left| z\right| \le 2}\frac{\left( z-1\right) ^{2}\mathrm{d}z}{\left( 1+t\left| \xi \right| ^{\frac{3}{2}}\left( z-1\right) ^{2}\right) ^{2}}+C\left| \xi \right| ^{\frac{5}{2}}\int _{\left| z\right| \ge 2}\frac{\left| z\right| ^{\frac{3}{2}}\mathrm{d}z}{1+t^{2}\left| \xi \right| ^{3}\left| z\right| ^{3}}\\&\le Ct^{-\frac{3}{2}}\left| \xi \right| ^{\frac{1}{4}}+Ct^{-\frac{5}{3}}. \end{aligned}$$

Lemma 2.2 is proved. $\square$

2.4 Estimates of pseudodifferential operators

There are many papers devoted to the $\mathbf {L}^{2}$-estimates of pseudodifferential operators (see, e.g., [2, 7, 8, 18]). Below, we will need the following result on the $\mathbf {L}^{2}$-boundedness of pseudodifferential operator $\mathbf {a}\left( t,x,\mathbf {D}\right) \phi \equiv \int _{\mathbb {R} }e^{ix\xi }\mathbf {a}\left( t,x,\xi \right) \widehat{\phi }\left( \xi \right) \mathrm{d}\xi$. See [1] for the proof.

Lemma 2.3

Let the symbol $\mathbf {a}\left( t,x,\xi \right)$ be such that $\sup _{x,\xi \in \mathbb {R},t\ge 1}\left| \frac{\left\langle \xi \right\rangle ^{\nu }}{\left\{ \xi \right\} ^{\nu }}\left( \xi \partial _{\xi }\right) ^{k}\mathbf {a}\left( t,x,\xi \right) \right| \le C$ for $k=0,1,2$, where $\nu \in \left( 0,1\right)$. Then $\left\| \mathbf {a} \left( t,x,\mathbf {D}\right) \phi \right\| _{_{\mathbf {L}_{x}^{2}}}\le C\left\| \phi \right\| _{_{\mathbf {L}^{2}}}$ for all $t\ge 1$.

Similarly, by considering the conjugate operator, we have.

Lemma 2.4

Let the symbol $\mathbf {a}\left( t,x,\xi \right)$ be such that $\sup _{x,\xi \in \mathbb {R},t\ge 1}\left| \left\{ x\right\} ^{-\nu }\left\langle x\right\rangle ^{\nu }\left( x\partial _{x}\right) ^{k}\mathbf {a}\left( t,x,\xi \right) \right| \le C$ for $k=0,1,2$, where $\nu \in \left( 0,1\right)$. Then $\left\| \mathbf {a}\left( t,x,\mathbf {D}\right) \phi \right\| _{_{\mathbf {L}_{x}^{2}}}\le C\left\| \phi \right\| _{_{\mathbf {L}^{2}}}$ for all $t\ge 1$.

2.5 Estimate for derivative of $\mathcal {Q}$

Next, we consider $\mathbf {L}^{2}$-estimate for the derivative $\partial _{\eta }\mathcal {Q}$.

Lemma 2.5

The estimate

$$\begin{aligned} \left\| \left| \Lambda ^{\prime \prime }\right| ^{\frac{1}{2}}\left\{ \eta \right\} ^{\frac{1}{2}-\nu }\partial _{\eta }\mathcal {Q}\phi \right\| _{\mathbf {L}^{2}\left( \left| \eta \right| \ge \frac{1}{3t}\right) }\le Ct^{2\nu }\left\| \xi \phi _{\xi }\right\| _{\mathbf {L}^{2} }+C\left\| \phi _{\xi }\right\| _{\mathbf {L}^{2}}+Ct^{\nu }\left\| \phi \right\| _{\mathbf {L}^{2}}+Ct^{\frac{2\nu }{1-4\nu }}\left\| \phi \right\| _{\mathbf {L}^{\infty }} \end{aligned}$$

is true for all $t\ge 1$, where $\nu >0$ is small.

Proof

Integrating by parts, we obtain

$$\begin{aligned} \partial _{\eta }\mathcal {Q}\phi&=-\sqrt{\frac{t}{2\pi }}\int _{\mathbb {R} }e^{-itS\left( \xi ,\eta \right) }\frac{\partial _{\eta }S\left( \xi ,\eta \right) }{\partial _{\xi }S\left( \xi ,\eta \right) }\phi _{\xi }\left( \xi \right) \mathrm{d}\xi \\&\quad -\sqrt{\frac{t}{2\pi }}\int _{\mathbb {R}}e^{-itS\left( \xi ,\eta \right) } \phi \left( \xi \right) \partial _{\xi }\left( \frac{\partial _{\eta }S\left( \xi ,\eta \right) }{\partial _{\xi }S\left( \xi ,\eta \right) }\right) \mathrm{d}\xi . \end{aligned}$$

Since $\partial _{\xi }S\left( \xi ,\eta \right) =\left| \xi \right| ^{-\frac{1}{2}}\xi -\left| \eta \right| ^{-\frac{1}{2}}\eta$ and $\partial _{\eta }S\left( \xi ,\eta \right) =\frac{1}{2}\left| \eta \right| ^{-\frac{1}{2}}\left( \eta -\xi \right)$, we get $-\frac{\partial _{\eta }S\left( \xi ,\eta \right) }{\partial _{\xi }S\left( \xi ,\eta \right) }=\frac{1}{2}\left| \eta \right| ^{-\frac{1}{2} }\left| \xi \right| ^{\frac{1}{2}}+\frac{1}{2}+b\left( \xi ,\eta \right)$, where $b\left( \xi ,\eta \right) =-\frac{\left| \xi \right| ^{\frac{1}{2}}}{\left| \xi \right| ^{\frac{1}{2} }+\left| \eta \right| ^{\frac{1}{2}}}\theta \left( -\xi \eta \right)$, $\theta \left( x\right)$ is the Heaviside function. Hence,

$$\begin{aligned} \partial _{\eta }\mathcal {Q}\phi&=\frac{1}{2}\left| \eta \right| ^{-\frac{1}{2}}\mathcal {Q}\left| \xi \right| ^{\frac{1}{2}}\phi _{\xi }+\frac{1}{2}\mathcal {Q}\phi _{\xi }+\frac{1}{4}\left| \eta \right| ^{-\frac{1}{2}}\mathcal {Q}\left| \xi \right| ^{\frac{1}{2}}\xi ^{-1} \phi \\&\quad +\sqrt{\frac{t}{2\pi }}\int _{\mathbb {R}}e^{-itS\left( \xi ,\eta \right) }\partial _{\xi }\left( b\left( \xi ,\eta \right) \phi \left( \xi \right) \right) \mathrm{d}\xi . \end{aligned}$$

Then, we represent

$$\begin{aligned} \partial _{\eta }\mathcal {Q}\phi&=\frac{1}{2}\left| \eta \right| ^{-\frac{1}{2}}\mathcal {Q}\left| \xi \right| ^{\frac{1}{2}}\phi _{\xi }+\frac{1}{2}\mathcal {Q}\phi _{\xi }\\&\quad +\frac{1}{4}\left| \eta \right| ^{-\frac{1}{2}}\mathcal {Q}\left| \xi \right| ^{\frac{1}{2}}\xi ^{-1}\left( \phi -\left\langle \xi \right\rangle ^{-1}\phi \left( 0\right) \right) +\frac{1}{4}\phi \left( 0\right) \left| \eta \right| ^{-\frac{1}{2}}\mathcal {Q}\left| \xi \right| ^{\frac{1}{2}}\xi ^{-1}\left\langle \xi \right\rangle ^{-1}\\&\quad +\sqrt{\frac{t}{2\pi }}\int _{\mathbb {R}}e^{-itS\left( \xi ,\eta \right) }b\left( \xi ,\eta \right) \phi _{\xi }\left( \xi \right) \mathrm{d}\xi \\&\quad +\sqrt{\frac{t}{2\pi }}\int _{\mathbb {R}}e^{-itS\left( \xi ,\eta \right) }\left\langle \xi \right\rangle ^{-\frac{1}{2}}\xi \partial _{\xi }b\left( \xi ,\eta \right) \frac{\left\langle \xi \right\rangle ^{\frac{1}{2}}\phi \left( \xi \right) -\phi \left( 0\right) }{\xi }\mathrm{d}\xi \\&\quad +\phi \left( 0\right) \sqrt{\frac{t}{2\pi }}\int _{\mathbb {R}}e^{-itS\left( \xi ,\eta \right) }\left\langle \xi \right\rangle ^{-\frac{1}{2}}\partial _{\xi }b\left( \xi ,\eta \right) \mathrm{d}\xi =\sum _{j=1}^{7}I_{j}. \end{aligned}$$

Note that $\left\| \left| \Lambda ^{\prime \prime }\right| ^{\frac{1}{2}}\mathcal {Q}\phi \right\| _{\mathbf {L}^{2}}=\left\| \phi \right\| _{\mathbf {L}^{2}}$. Hence using inequality $t^{\nu }\left| \xi \right| ^{\frac{1}{2}}\le 1+t^{2\nu }\left| \xi \right|$, we get

$$\begin{aligned} \left\| \left| \Lambda ^{\prime \prime }\right| ^{\frac{1}{2} }\left\{ \eta \right\} ^{\frac{1}{2}-\nu }I_{1}\right\| _{\mathbf {L} ^{2}\left( \left| \eta \right| \ge \frac{1}{3t}\right) }&\le Ct^{\nu }\left\| \left| \Lambda ^{\prime \prime }\right| ^{\frac{1}{2} }\mathcal {Q}\left| \xi \right| ^{\frac{1}{2}}\phi _{\xi }\right\| _{\mathbf {L}^{2}}\\&=Ct^{\nu }\left\| \left| \xi \right| ^{\frac{1}{2}}\phi _{\xi }\right\| _{\mathbf {L}^{2}}\le C\left\| \phi _{\xi }\right\| _{\mathbf {L}^{2}}+Ct^{2\nu }\left\| \xi \phi _{\xi }\right\| _{\mathbf {L} ^{2}}. \end{aligned}$$

Similarly $\left\| \left| \Lambda ^{\prime \prime }\right| ^{\frac{1}{2}}\left\{ \eta \right\} ^{\frac{1}{2}-\nu }I_{2}\right\| _{\mathbf {L}^{2}\left( \left| \eta \right| \ge \frac{1}{3t}\right) }\le C\left\| \left| \Lambda ^{\prime \prime }\right| ^{\frac{1}{2} }\mathcal {Q}\phi _{\xi }\right\| _{\mathbf {L}^{2}}\le C\left\| \phi _{\xi }\right\| _{\mathbf {L}^{2}}$. Also we find

$$\begin{aligned} \left\| \left| \Lambda ^{\prime \prime }\right| ^{\frac{1}{2} }\left\{ \eta \right\} ^{\frac{1}{2}-\nu }I_{3}\right\| _{\mathbf {L} ^{2}\left( \left| \eta \right| \ge \frac{1}{3t}\right) }&\le Ct^{\nu }\left\| \left| \Lambda ^{\prime \prime }\right| ^{\frac{1}{2} }\mathcal {Q}\left| \xi \right| ^{\frac{1}{2}}\xi ^{-1}\left( \phi -\left\langle \xi \right\rangle ^{-1}\phi \left( 0\right) \right) \right\| _{\mathbf {L}^{2}}\\&\le Ct^{\nu }\left\| \left| \xi \right| ^{-\frac{1}{2}}\left( \phi -\left\langle \xi \right\rangle ^{-1}\phi \left( 0\right) \right) \right\| _{\mathbf {L}^{2}}\\&\le C\left\| \partial _{\xi }\phi \right\| _{\mathbf {L}^{2}}+Ct^{\nu }\left\| \phi \right\| _{\mathbf {L}^{2}}+Ct^{\nu }\log t\left\| \phi \right\| _{\mathbf {L}^{\infty }}, \end{aligned}$$

since

$$\begin{aligned} \left\| \left| \xi \right| ^{-\frac{1}{2}}\left( \phi -\left\langle \xi \right\rangle ^{-1}\phi \left( 0\right) \right) \right\| _{\mathbf {L}^{2}}^{2}&\le C\int _{0}^{t^{-2\nu }}\left( \phi \left( \xi \right) -\left\langle \xi \right\rangle ^{-1}\phi \left( 0\right) \right) ^{2} \frac{\mathrm{d}\xi }{\xi }+C\left\| \phi \right\| _{\mathbf {L}^{\infty }} \int _{t^{-2\nu }}^{1}\frac{\mathrm{d}\xi }{\xi }\\&\quad +C\int _{1}^{\infty }\left( \left| \phi \left( \xi \right) \right| ^{2}+\left\langle \xi \right\rangle ^{-2}\left| \phi \left( 0\right) \right| ^{2}\right) \frac{\mathrm{d}\xi }{\xi }\\&\le Ct^{-2\nu }\left\| \partial _{\xi }\phi \right\| _{\mathbf {L}^{2}}^{2}+C\log t\left\| \phi \right\| _{\mathbf {L}^{\infty }}^{2}+C\left\| \phi \right\| _{\mathbf {L}^{2}}^{2}. \end{aligned}$$

Since $\left\| \mathcal {Q}\phi \right\| _{\mathbf {L}^{\infty }}\le C\left| t\right| ^{\frac{1}{2}}\left\| \phi \right\| _{\mathbf {L}^{1}}$, then by the Riesz interpolation theorem (see [29], p. 52), we have $\left\| \left| \Lambda ^{\prime \prime }\right| ^{\frac{1}{p}}\mathcal {Q}\left( t\right) \phi \right\| _{\mathbf {L}^{p} }\le C\left| t\right| ^{\frac{1}{2}-\frac{1}{p}}\left\| \phi \right\| _{\mathbf {L}^{\frac{p}{p-1}}}$ for $2\le p\le \infty$. Hence taking $p=2+\frac{8\nu }{1-8\nu }$, we find

$$\begin{aligned}&\left\| \left| \Lambda ^{\prime \prime }\right| ^{\frac{1}{2} }\left\{ \eta \right\} ^{\frac{1}{2}-\nu }I_{4}\right\| _{\mathbf {L} ^{2}\left( \left| \eta \right| \ge \frac{1}{3t}\right) }\\&\quad \le C\left| \phi \left( 0\right) \right| \left\| \left| \Lambda ^{\prime \prime }\right| ^{\frac{1}{2}-\frac{1}{p}}\left\{ \eta \right\} ^{-\nu }\left\langle \eta \right\rangle ^{-\frac{1}{2}}\right\| _{\mathbf {L}^{\frac{2p}{p-2}}\left( \left| \eta \right| \ge \frac{1}{3t}\right) }\left\| \left| \Lambda ^{\prime \prime }\right| ^{\frac{1}{p}}\mathcal {Q}\left| \xi \right| ^{\frac{1}{2}}\xi ^{-1}\left\langle \xi \right\rangle ^{-1}\right\| _{\mathbf {L}^{p}}\\&\quad \le C\left| \phi \left( 0\right) \right| \left| t\right| ^{\frac{1}{2}-\frac{1}{p}}\left\| \left| \xi \right| ^{\frac{1}{2} }\xi ^{-1}\left\langle \xi \right\rangle ^{-1}\right\| _{\mathbf {L}^{\frac{p}{p-1}}}\le C\left| t\right| ^{\frac{2\nu }{1-4\nu }}\left\| \phi \right\| _{\mathbf {L}^{\infty }} \end{aligned}$$

since $\left( \frac{1}{2}\left( \frac{1}{2}-\frac{1}{p}\right) +\nu \right) \frac{2p}{p-2}<1$, in view of $p>2+\frac{8\nu }{1-4\nu }$. Next, we change $\eta =\mu \left( x\right)$, then we get

$$\begin{aligned} \left\{ \eta \right\} ^{\frac{1}{2}-\nu }I_{5}=\sqrt{\frac{1}{2\pi }} \overline{M}\mathcal {B}^{-1}\mathcal {D}_{t}^{-1}\int _{\mathbb {R}}e^{ix\xi }b_{1}\left( \mu \left( xt^{-1}\right) ,\xi \right) e^{-it\Lambda \left( \xi \right) }\left\langle \xi \right\rangle ^{\frac{1}{2}}\phi _{\xi }\left( \xi \right) \mathrm{d}\xi , \end{aligned}$$

where $b_{1}\left( \xi ,\eta \right) =\left\{ \eta \right\} ^{\frac{1}{2} -\nu }\left\langle \xi \right\rangle ^{-\frac{1}{2}}b\left( \xi ,\eta \right)$, and similarly

$$\begin{aligned} \left\{ \eta \right\} ^{\frac{1}{2}-\nu }I_{6}=\sqrt{\frac{1}{2\pi }} \overline{M}\mathcal {B}^{-1}\mathcal {D}_{t}^{-1}\int _{\mathbb {R}}e^{ix\xi }b_{2}\left( \mu \left( xt^{-1}\right) ,\xi \right) e^{-it\Lambda \left( \xi \right) }\frac{\left\langle \xi \right\rangle ^{\frac{1}{2}}\phi \left( \xi \right) -\phi \left( 0\right) }{\xi }\mathrm{d}\xi , \end{aligned}$$

where $b_{2}\left( \xi ,\eta \right) =\left\{ \eta \right\} ^{\frac{1}{2} -\nu }\left\langle \xi \right\rangle ^{-\frac{1}{2}}\xi \partial _{\xi }b\left( \xi ,\eta \right)$. Define the pseudodifferential operators $\mathbf {a} _{k}\left( t,x,D\right) \phi \equiv \int _{\mathbb {R}}e^{ix\xi }\mathbf {a} _{k}\left( t,x,\xi \right) \widehat{\phi }\left( \xi \right) \mathrm{d}\xi$ with symbols $\mathbf {a}_{k}\left( t,x,\xi \right) =\sqrt{\frac{1}{2\pi }} b_{k}\left( \mu \left( xt^{-1}\right) ,\xi \right)$. Then, we get

$$\begin{aligned} \left\{ \eta \right\} ^{\frac{1}{2}-\nu }I_{5}&=\overline{M}\mathcal {B} ^{-1}\mathcal {D}_{t}^{-1}\mathbf {a}_{1}\left( t,x,\mathbf {D}\right) \mathcal {F}^{-1}e^{-it\Lambda }\left\langle \xi \right\rangle ^{\frac{1}{2}} \phi _{\xi },\\ \left\{ \eta \right\} ^{\frac{1}{2}-\nu }I_{6}&=\overline{M}\mathcal {B} ^{-1}\mathcal {D}_{t}^{-1}\mathbf {a}_{2}\left( t,x,\mathbf {D}\right) \mathcal {F}^{-1}e^{-it\Lambda }\frac{\left\langle \xi \right\rangle ^{\frac{1}{2}}\phi \left( \xi \right) -\phi \left( 0\right) }{\xi }. \end{aligned}$$

Let us prove the $\mathbf {L}^{2}$-boundedness of the pseudodifferential operators $\mathbf {a}_{k}\left( t,x,D\right)$. We have

$$\begin{aligned} \mathbf {a}_{k}\left( t,x,\xi \right) =-\sqrt{\frac{1}{2\pi }}\left. \left\langle \xi \right\rangle ^{-\frac{1}{2}}\left( \xi \partial _{\xi }\right) ^{k-1}\left( \frac{\left\{ \eta \right\} ^{\frac{1}{2}-\nu }\left| \xi \right| ^{\frac{1}{2}}}{\left| \xi \right| ^{\frac{1}{2} }+\left| \eta \right| ^{\frac{1}{2}}}\theta \left( -\xi \eta \right) \right) \right| _{\eta =\mu \left( xt^{-1}\right) }. \end{aligned}$$

Hence, we obtain the estimates

$$\begin{aligned}&\left| \frac{\left\langle \xi \right\rangle ^{\nu }}{\left\{ \xi \right\} ^{\nu }}\left( \xi \partial _{\xi }\right) ^{l}\mathbf {a}_{k}\left( t,x,\xi \right) \right| \\&=O\left( \left. \frac{\left\langle \xi \right\rangle ^{\nu }}{\left\{ \xi \right\} ^{\nu }}\left( \xi \partial _{\xi }\right) ^{l}\left\langle \xi \right\rangle ^{-\frac{1}{2}}\left( \xi \partial _{\xi }\right) ^{k-1}\left( \frac{\left\{ \eta \right\} ^{\frac{1}{2}-\nu }\left| \xi \right| ^{\frac{1}{2}}}{\left| \xi \right| ^{\frac{1}{2} }+\left| \eta \right| ^{\frac{1}{2}}}\theta \left( -\xi \eta \right) \right) \right| _{\eta =\mu \left( xt^{-1}\right) }\right) \le C \end{aligned}$$

for all $x,\xi \in \mathbb {R}$, $t\ge 1,~k=0,1,2$, $k=1,2$, with $\nu >0$. Therefore, by Lemma 2.3, we find $\left\| \mathbf {a}_{k}\left( t,x,\mathbf {D}\right) \phi \right\| _{_{\mathbf {L}_{x}^{2}}}\le C\left\| \phi \right\| _{_{\mathbf {L}^{2}}}$, $k=1,2$. Then, we obtain

$$\begin{aligned} \left\| \left| \Lambda ^{\prime \prime }\right| ^{\frac{1}{2} }\left\{ \eta \right\} ^{\frac{1}{2}-\nu }I_{5}\right\| _{\mathbf {L}_{\eta }^{2}}&\le \left\| \mathbf {a}_{1}\left( t,x,\mathbf {D}\right) \mathcal {F}^{-1}e^{-it\Lambda }\left\langle \xi \right\rangle ^{\frac{1}{2}} \phi _{\xi }\right\| _{\mathbf {L}_{x}^{2}}\\&\le C\left\| \mathcal {F}^{-1}e^{-it\Lambda }\left\langle \xi \right\rangle ^{\frac{1}{2}}\phi _{\xi }\right\| _{\mathbf {L}^{2}}\le C\left\| \left\langle \xi \right\rangle ^{\frac{1}{2}}\phi _{\xi }\right\| _{\mathbf {L}^{2}} \end{aligned}$$

and

$$\begin{aligned} \left\| \left| \Lambda ^{\prime \prime }\right| ^{\frac{1}{2} }\left\{ \eta \right\} ^{\frac{1}{2}-\nu }I_{6}\right\| _{\mathbf {L}_{\eta }^{2}}&\le \left\| \mathbf {a}_{2}\left( t,x,\mathbf {D}\right) \mathcal {F}^{-1}e^{-it\Lambda }\frac{\left\langle \xi \right\rangle ^{\frac{1}{2}}\phi \left( \xi \right) -\phi \left( 0\right) }{\xi }\right\| _{\mathbf {L}_{x}^{2}}\\&\le C\left\| \frac{\left\langle \xi \right\rangle ^{\frac{1}{2}} \phi \left( \xi \right) -\phi \left( 0\right) }{\xi }\right\| _{\mathbf {L}^{2}}\le C\left\| \left\langle \xi \right\rangle ^{\frac{1}{2} }\phi _{\xi }\right\| _{\mathbf {L}^{2}}+C\left\| \phi \right\| _{\mathbf {L}^{2}}. \end{aligned}$$

Finally, we need to estimate the integral $I_{7}$. Consider $\eta >0$, then we get

$$\begin{aligned} \left\{ \eta \right\} ^{\frac{1}{2}-\nu }I_{7}=-\phi \left( 0\right) \sqrt{\frac{t}{2\pi }}\int _{-\infty }^{0}e^{-itS\left( \xi ,\eta \right) }\left\langle \xi \right\rangle ^{-\frac{1}{2}}\partial _{\xi }\frac{\left\{ \eta \right\} ^{\frac{1}{2}-\nu }\left| \xi \right| ^{\frac{1}{2}} }{\left| \xi \right| ^{\frac{1}{2}}+\left| \eta \right| ^{\frac{1}{2}}}\mathrm{d}\xi . \end{aligned}$$

We integrate by parts via the identity $e^{-itS\left( \xi ,\eta \right) }=H_{4}\partial _{\xi }\left( \xi e^{-itS\left( \xi ,\eta \right) }\right)$ with $H_{4}=\left( 1-it\xi \partial _{\xi }S\left( \xi ,\eta \right) \right) ^{-1}$ with $\partial _{\xi }S\left( \xi ,\eta \right) =\left| \xi \right| ^{-\frac{1}{2}}\xi -\left| \eta \right| ^{-\frac{1}{2} }\eta =-\left( \left| \xi \right| ^{\frac{1}{2}}+\left| \eta \right| ^{\frac{1}{2}}\right)$ for $\eta >0$, $\xi <0$. Therefore,

$$\begin{aligned} \left\{ \eta \right\} ^{\frac{1}{2}-\nu }I_{7}=Ct^{\frac{1}{2}}\phi \left( 0\right) \int _{-\infty }^{0}e^{-itS\left( \xi ,\eta \right) }\xi \partial _{\xi }\left( H_{4}\left\langle \xi \right\rangle ^{-\frac{1}{2}}\partial _{\xi } \frac{\left\{ \eta \right\} ^{\frac{1}{2}-\nu }\left| \xi \right| ^{\frac{1}{2}}}{\left| \xi \right| ^{\frac{1}{2}}+\left| \eta \right| ^{\frac{1}{2}}}\right) \mathrm{d}\xi . \end{aligned}$$

We have

$$\begin{aligned} \left| \xi \partial _{\xi }\left( H_{4}\left\langle \xi \right\rangle ^{-\frac{1}{2}}\partial _{\xi }\frac{\left\{ \eta \right\} ^{\frac{1}{2}-\nu }\left| \xi \right| ^{\frac{1}{2}}}{\left| \xi \right| ^{\frac{1}{2}}+\left| \eta \right| ^{\frac{1}{2}}}\right) \right| \le \frac{C\left\langle \xi \right\rangle ^{-\frac{1}{2}}\left\{ \eta \right\} ^{\frac{1}{2}-\nu }\left| \xi \right| ^{-\frac{1}{2}}}{\left( \left| \xi \right| ^{\frac{1}{2}}+\left| \eta \right| ^{\frac{1}{2}}\right) \left( 1+t\left| \xi \right| \left( \left| \xi \right| ^{\frac{1}{2}}+\left| \eta \right| ^{\frac{1}{2} }\right) \right) }. \end{aligned}$$

Hence changing $\xi =\eta z$, we get

$$\begin{aligned} \left\{ \eta \right\} ^{\frac{1}{2}-\nu }I_{7}&\le Ct^{\frac{1}{2} }\left| \phi \left( 0\right) \right| \int _{-\infty }^{0} \frac{\left\langle \xi \right\rangle ^{-\frac{1}{2}}\left\{ \eta \right\} ^{\frac{1}{2}-\nu }\left| \xi \right| ^{-\frac{1}{2}}\mathrm{d}\xi }{\left( \left| \xi \right| ^{\frac{1}{2}}+\left| \eta \right| ^{\frac{1}{2}}\right) \left( 1+t\left| \xi \right| \left( \left| \xi \right| ^{\frac{1}{2}}+\left| \eta \right| ^{\frac{1}{2} }\right) \right) }\\&\le Ct^{\frac{1}{2}}\left\{ \eta \right\} ^{\frac{1}{2}-\nu }\left| \phi \left( 0\right) \right| \int _{-\infty }^{0}\frac{\mathrm{d}z}{\left| z\right| ^{\frac{1}{2}}\left( 1+t\left| \eta \right| ^{\frac{3}{2}}\left| z\right| \right) }\\&\le Ct^{\frac{1}{2}}\left\{ \eta \right\} ^{\frac{1}{2}-\nu }\left| \phi \left( 0\right) \right| \left\langle t\left| \eta \right| ^{\frac{3}{2}}\right\rangle ^{-\frac{1}{2}}. \end{aligned}$$

Therefore,

$$\begin{aligned} \left\| \left| \Lambda ^{\prime \prime }\right| ^{\frac{1}{2} }\left\{ \eta \right\} ^{\frac{1}{2}-\nu }I_{7}\right\| _{\mathbf {L}_{\eta }^{2}}&\le Ct^{\frac{1}{2}}\left| \phi \left( 0\right) \right| \left\| \frac{\left\{ \eta \right\} ^{\frac{1}{4}-\nu }}{\left\langle t\left| \eta \right| ^{\frac{3}{2}}\right\rangle ^{\frac{1}{2}} }\right\| _{\mathbf {L}_{\eta }^{2}}\\&\le Ct^{\frac{1}{2}}\left| \phi \left( 0\right) \right| \left( \int _{0}^{1}\frac{\eta ^{\frac{1}{2}-2\nu }}{1+t\left| \eta \right| ^{\frac{3}{2}}}\mathrm{d}\eta \right) ^{\frac{1}{2}}+Ct^{\frac{1}{2}}\left| \phi \left( 0\right) \right| \left( \int _{1}^{\infty }\frac{\mathrm{d}\eta }{t\left| \eta \right| ^{\frac{3}{2}}}\right) ^{\frac{1}{2}}\\&\le Ct^{\frac{2}{3}\nu }\left| \phi \left( 0\right) \right| \left( \int _{0}^{t^{\frac{2}{3}}}\frac{\eta ^{\frac{1}{2}-2\nu }\mathrm{d}\eta }{1+\left| \eta \right| ^{\frac{3}{2}}}\right) ^{\frac{1}{2}}+C\left| \phi \left( 0\right) \right| \le Ct^{\frac{2}{3}\nu }\left| \phi \left( 0\right) \right| . \end{aligned}$$

Lemma 2.5 is proved. $\square$

2.6 Estimate for derivative of $\mathcal {Q}^{*}$

In the next lemma, we estimate the derivative $\partial _{\xi }\mathcal {Q}^{*}$ in the domain $\left| \xi \right| \le 1$.

Lemma 2.6

The estimate

$$\begin{aligned} \left\| \partial _{\xi }\mathcal {Q}^{*}\phi \right\| _{\mathbf {L} ^{2}\left( \left| \xi \right| \le 1\right) }&\le C\left\| \left| \Lambda ^{\prime \prime }\right| ^{\frac{1}{2}}\left\{ \eta \right\} ^{-\nu }\partial _{\eta }\phi \right\| _{\mathbf {L}^{2}\left( \left| \eta \right| \ge \frac{1}{3t}\right) }+C\left\| \left| \Lambda ^{\prime \prime }\right| ^{\frac{1}{2}}\left\{ \eta \right\} ^{-1-\nu }\phi \right\| _{\mathbf {L}^{2}\left( \left| \eta \right| \ge \frac{1}{3t}\right) }\\&\quad +Ct^{\frac{3}{4}}\left\| \phi \right\| _{\mathbf {L}^{\infty }\left( \left| \eta \right| \le t^{-1}\right) } \end{aligned}$$

is true for all $t\ge 1$, where $\nu >0$ is small.

Proof

Using $\chi _{1}\left( x\right) \in \mathbf {C}^{4}\left( \mathbb {R}\right)$ such that $\chi _{1}\left( x\right) =1$ for $\left| x\right| \le \frac{1}{3}$ and $\chi _{1}\left( x\right) =0$ for $\left| x\right| \ge \frac{2}{3}$, $\chi _{2}\left( x\right) =1-\chi _{1}\left( x\right)$, we write

$$\begin{aligned} \partial _{\xi }\mathcal {Q}^{*}\phi&=it\sqrt{\frac{t}{2\pi }}\int _{\mathbb {R}}e^{itS\left( \xi ,\eta \right) }\partial _{\xi }S\left( \xi ,\eta \right) \phi \left( \eta \right) \chi _{1}\left( \eta t\right) \Lambda ^{\prime \prime }\left( \eta \right) \mathrm{d}\eta \\&\quad +it\sqrt{\frac{t}{2\pi }}\int _{\mathbb {R}}e^{itS\left( \xi ,\eta \right) }\partial _{\xi }S\left( \xi ,\eta \right) \chi _{2}\left( \eta t\right) \phi \left( \eta \right) \Lambda ^{\prime \prime }\left( \eta \right) \mathrm{d}\eta =I_{1}+I_{2}. \end{aligned}$$

Note that $\left\| \mathcal {Q}^{*}\phi \right\| _{\mathbf {L}^{2} }=\left\| \left| \Lambda ^{\prime \prime }\right| ^{\frac{1}{2}} \phi \right\| _{\mathbf {L}^{2}}$. Then, the first integral, we can estimate as follows

$$\begin{aligned} \left\| I_{1}\right\| _{\mathbf {L}^{2}\left( \left| \xi \right| \le 1\right) }&\le Ct\left\| \Lambda ^{\prime } \mathcal {Q}^{*}\chi _{1}\left( \eta t\right) \phi \right\| _{\mathbf {L}^{2}\left( \left| \xi \right| \le 1\right) }+Ct\left\| \mathcal {Q}^{*}\Lambda ^{\prime }\chi _{1}\left( \eta t\right) \phi \right\| _{\mathbf {L}^{2}\left( \left| \xi \right| \le 1\right) }\\&\le Ct\left\| \left| \Lambda ^{\prime \prime }\right| ^{\frac{1}{2}}\phi \right\| _{\mathbf {L}^{2}\left( \left| \eta \right| \le t^{-1}\right) }\le Ct\left\| \phi \right\| _{\mathbf {L}^{\infty }\left( \left| \eta \right| \le t^{-1}\right) }\left\| \left| \eta \right| ^{-\frac{1}{4}}\right\| _{\mathbf {L}^{2}\left( \left| \eta \right| \le t^{-1}\right) }\\&\le Ct^{\frac{3}{4}}\left\| \phi \right\| _{\mathbf {L}^{\infty }\left( \left| \eta \right| \le t^{-1}\right) }. \end{aligned}$$

Integrating by parts, we obtain for the second integral

$$\begin{aligned} I_{2}&=-\sqrt{\frac{t}{2\pi }}\int _{\mathbb {R}}e^{itS\left( \xi ,\eta \right) }\frac{\partial _{\xi }S\left( \xi ,\eta \right) }{\partial _{\eta }S\left( \xi ,\eta \right) }t\chi _{2}^{\prime }\left( \eta t\right) \phi \left( \eta \right) \Lambda ^{\prime \prime }\left( \eta \right) \mathrm{d}\eta \\&\quad -\sqrt{\frac{t}{2\pi }}\int _{\mathbb {R}}e^{itS\left( \xi ,\eta \right) } \frac{\partial _{\xi }S\left( \xi ,\eta \right) }{\partial _{\eta }S\left( \xi ,\eta \right) }\chi _{2}\left( \eta t\right) \partial _{\eta }\phi \left( \eta \right) \Lambda ^{\prime \prime }\left( \eta \right) \mathrm{d}\eta \\&\quad -\sqrt{\frac{t}{2\pi }}\int _{\mathbb {R}}e^{itS\left( \xi ,\eta \right) } \chi _{2}\left( \eta t\right) \phi \left( \eta \right) \partial _{\eta }\left( \frac{\Lambda ^{\prime \prime }\left( \eta \right) \partial _{\xi }S\left( \xi ,\eta \right) }{\partial _{\eta }S\left( \xi ,\eta \right) }\right) \mathrm{d}\eta =\sum _{j=3}^{5}I_{j}. \end{aligned}$$

Since $\partial _{\xi }S\left( \xi ,\eta \right) =\left| \xi \right| ^{-\frac{1}{2}}\xi -\left| \eta \right| ^{-\frac{1}{2}}\eta$ and $\partial _{\eta }S\left( \xi ,\eta \right) =\Lambda ^{\prime \prime }\left( \eta \right) \left( \eta -\xi \right)$, we get $-\frac{\partial _{\xi }S\left( \xi ,\eta \right) }{\partial _{\eta }S\left( \xi ,\eta \right) }=2+B\left( \xi ,\eta \right)$, where $B\left( \xi ,\eta \right) =-\frac{2\left| \xi \right| ^{\frac{1}{2}}}{\left| \xi \right| ^{\frac{1}{2} }+\left| \eta \right| ^{\frac{1}{2}}}\theta \left( \xi \eta \right) +\frac{2\left| \xi \right| ^{\frac{1}{2}}\left( \left| \eta \right| ^{\frac{1}{2}}-\left| \xi \right| ^{\frac{1}{2} }\right) }{\left| \xi \right| +\left| \eta \right| } \theta \left( -\xi \eta \right)$. Also, we find $-\frac{1}{\Lambda ^{\prime \prime }\left( \eta \right) }\partial _{\eta }\frac{\Lambda ^{\prime \prime }\left( \eta \right) \partial _{\xi }S\left( \xi ,\eta \right) }{\partial _{\eta }S\left( \xi ,\eta \right) }=-\frac{1}{4}+B_{1}\left( \xi ,\eta \right)$, where $B_{1}\left( \xi ,\eta \right) =\frac{\frac{1}{4}\left| \xi \right| ^{\frac{1}{2}}\left( 2\left| \eta \right| ^{\frac{1}{2}}+\left| \xi \right| ^{\frac{1}{2}}\right) }{\left( \left| \xi \right| ^{\frac{1}{2}}+\left| \eta \right| ^{\frac{1}{2}}\right) ^{2}}\theta \left( \xi \eta \right) -\frac{\frac{1}{2}\left| \xi \right| ^{\frac{1}{2}}\left( \left| \eta \right| ^{\frac{3}{2}}-\frac{1}{2}\left| \xi \right| ^{\frac{3}{2}}-\frac{3}{2}\left| \xi \right| ^{\frac{1}{2}}\left| \eta \right| \right) }{\left( \left| \xi \right| +\left| \eta \right| \right) ^{2} }\theta \left( -\xi \eta \right)$. Hence, we represent $I_{4}=2\mathcal {Q} ^{*}\chi _{2}\left( \eta t\right) \partial _{\eta }\phi \left( \eta \right) +I_{6}$ and $I_{5}=-\frac{1}{4}\mathcal {Q}^{*}\chi _{2}\left( \eta t\right) \frac{\phi \left( \eta \right) }{\eta }+I_{7}$ , where

$$\begin{aligned} I_{6}&=\sqrt{\frac{t}{2\pi }}\int _{\mathbb {R}}e^{itS\left( \xi ,\eta \right) }B\left( \xi ,\eta \right) \chi _{2}\left( \eta t\right) \partial _{\eta } \phi \left( \eta \right) \Lambda ^{\prime \prime }\left( \eta \right) \mathrm{d}\eta ,\\ I_{7}&=\sqrt{\frac{t}{2\pi }}\int _{\mathbb {R}}e^{itS\left( \xi ,\eta \right) }B_{1}\left( \xi ,\eta \right) \chi _{2}\left( \eta t\right) \frac{\phi \left( \eta \right) }{\eta }\Lambda ^{\prime \prime }\left( \eta \right) \mathrm{d}\eta . \end{aligned}$$

The integral $I_{3}$ is estimated as $I_{1}$ above

$$\begin{aligned} \left\| I_{3}\right\| _{\mathbf {L}^{2}\left( \left| \xi \right| \le 1\right) }\le Ct\left\| \phi \right\| _{\mathbf {L}^{\infty }\left( \left| \eta \right| \le t^{-1}\right) }\left\| \left| \eta \right| ^{-\frac{1}{4}}\right\| _{\mathbf {L}^{2}\left( \left| \eta \right| \le t^{-1}\right) }\le Ct^{\frac{3}{4}}\left\| \phi \right\| _{\mathbf {L}^{\infty }\left( \left| \eta \right| \le t^{-1}\right) }. \end{aligned}$$

Also, we find

$$\begin{aligned} \left\| \mathcal {Q}^{*}\chi _{2}\left( \eta t\right) \partial _{\eta }\phi \left( \eta \right) \right\| _{\mathbf {L}^{2}\left( \left| \xi \right| \le 1\right) }\le \left\| \left| \Lambda ^{\prime \prime }\right| ^{\frac{1}{2}}\partial _{\eta }\phi \right\| _{\mathbf {L} ^{2}\left( \left| \eta \right| \ge \frac{1}{3t}\right) } \end{aligned}$$

and

$$\begin{aligned} \left\| \mathcal {Q}^{*}\chi _{2}\left( \eta t\right) \frac{\phi \left( \eta \right) }{\eta }\right\| _{\mathbf {L}^{2}}\le \left\| \left| \Lambda ^{\prime \prime }\right| ^{\frac{1}{2}}\frac{\phi \left( \eta \right) }{\eta }\right\| _{\mathbf {L}^{2}\left( \left| \eta \right| \ge \frac{1}{3t}\right) }. \end{aligned}$$

Then changing the variable of integration $\eta =\mu \left( x\right)$, we get

$$\begin{aligned}&\chi _{1}\left( 3\xi \right) I_{6}=e^{it\Lambda \left( \xi \right) } \int _{\mathbb {R}}e^{-ix\xi }b_{1}^{*}\left( \xi ,\mu \left( xt^{-1}\right) \right) \left( \mathcal {D}_{t}\mathcal {B}M\left\{ \eta \right\} ^{-\nu } \chi _{2}\left( \eta t\right) \partial _{\eta }\phi \right) \mathrm{d}x,\\&\chi _{1}\left( 3\xi \right) I_{7}=e^{it\Lambda \left( \xi \right) } \int _{\mathbb {R}}e^{-ix\xi }b_{2}^{*}\left( \xi ,\mu \left( xt^{-1}\right) \right) \left( \mathcal {D}_{t}\mathcal {B}M\left\{ \eta \right\} ^{-\nu } \chi _{2}\left( \eta t\right) \frac{\phi \left( \eta \right) }{\eta }\right) \mathrm{d}x, \end{aligned}$$

where $b_{1}^{*}\left( \xi ,\eta \right) =\sqrt{\frac{t}{2\pi }}\chi _{1}\left( 3\xi \right) \left\{ \eta \right\} ^{\nu }B\left( \xi ,\eta \right)$ and $b_{2}^{*}\left( \xi ,\eta \right) =\sqrt{\frac{t}{2\pi }}\chi _{1}\left( 3\xi \right) \left\{ \eta \right\} ^{\nu }B_{1}\left( \xi ,\eta \right)$. We define the pseudodifferential operators $\mathbf {a} _{k}^{*}\left( t,\xi ,\mathbf {D}\right) \phi =\int _{\mathbb {R}}e^{-ix\xi }\mathbf {a}_{k}\left( t,x,\xi \right) \widehat{\phi }\left( x\right) \mathrm{d}x$, with symbols $\mathbf {a}_{k}\left( t,x,\xi \right) =b_{k}^{*}\left( \xi ,\mu \left( xt^{-1}\right) \right)$, and then we get

$$\begin{aligned} \chi _{1}\left( 3\xi \right) I_{6}&=e^{it\Lambda \left( \xi \right) }\mathbf {a}_{1}^{*}\left( t,\xi ,\mathbf {D}\right) \mathcal {F} ^{-1}\mathcal {D}_{t}\mathcal {B}M\left\{ \eta \right\} ^{-\nu }\chi _{2}\left( \eta t\right) \partial _{\eta }\phi \left( \eta \right) ,\\ \chi _{1}\left( 3\xi \right) I_{7}&=e^{it\Lambda \left( \xi \right) }\mathbf {a}_{2}^{*}\left( t,\xi ,\mathbf {D}\right) \mathcal {F} ^{-1}\mathcal {D}_{t}\mathcal {B}M\chi _{2}\left( \eta t\right) \frac{\phi \left( \eta \right) }{\eta \left\{ \eta \right\} ^{\nu }}. \end{aligned}$$

We prove the $\mathbf {L}^{2}$-boundedness of the pseudodifferential operators $\mathbf {a}_{k}^{*}\left( t,\xi ,\mathbf {D}\right)$, $k=1,2$. We have

$$\begin{aligned} \mathbf {a}_{1}\left( t,x,\xi \right)&= \left. -\sqrt{\frac{t}{2\pi }} \frac{2\chi _{1}\left( 3\xi \right) \left| \xi \right| ^{\frac{1}{2} }\left\{ \eta \right\} ^{\nu }}{\left| \xi \right| ^{\frac{1}{2} }+\left| \eta \right| ^{\frac{1}{2}}}\theta \left( \xi \eta \right) \right| _{\eta =\mu \left( xt^{-1}\right) }\\&=\left. \sqrt{\frac{t}{2\pi }}\frac{2\chi _{1}\left( 3\xi \right) \left\{ \eta \right\} ^{\nu }\left| \xi \right| ^{\frac{1}{2}}\left( \left| \eta \right| ^{\frac{1}{2}}-\left| \xi \right| ^{\frac{1}{2} }\right) }{\left| \xi \right| +\left| \eta \right| } \theta \left( -\xi \eta \right) \right| _{\eta =\mu \left( xt^{-1}\right) } \end{aligned}$$

and

$$\begin{aligned} \mathbf {a}_{2}\left( t,x,\xi \right)&=\left. \frac{1}{4}\sqrt{\frac{t}{2\pi }}\frac{\chi _{1}\left( 3\xi \right) \left\{ \eta \right\} ^{\nu }\left| \xi \right| ^{\frac{1}{2}}\left( 2\left| \eta \right| ^{\frac{1}{2}}+\left| \xi \right| ^{\frac{1}{2}}\right) }{\left( \left| \xi \right| ^{\frac{1}{2}}+\left| \eta \right| ^{\frac{1}{2}}\right) ^{2}}\theta \left( \xi \eta \right) \right| _{\eta =\mu \left( xt^{-1}\right) }\\&\quad -\left. \frac{1}{2}\sqrt{\frac{t}{2\pi }}\frac{\chi _{1}\left( 3\xi \right) \left\{ \eta \right\} ^{\nu }\left| \xi \right| ^{\frac{1}{2}}\left( \left| \eta \right| ^{\frac{3}{2}}-\frac{1}{2}\left| \xi \right| ^{\frac{3}{2}}-\frac{3}{2}\left| \xi \right| ^{\frac{1}{2}}\left| \eta \right| \right) }{\left( \left| \xi \right| +\left| \eta \right| \right) ^{2}}\theta \left( -\xi \eta \right) \right| _{\eta =\mu \left( xt^{-1}\right) }. \end{aligned}$$

Then, we obtain the estimates

$$\begin{aligned}&\left| \left\langle \xi \right\rangle ^{\nu }\left\{ \xi \right\} ^{-\nu }\left( \xi \partial _{\xi }\right) ^{l}\mathbf {a}_{1}\left( t,x,\xi \right) \right| \\&\quad =O\left( \left. \left\{ \xi \right\} ^{-\nu }\left( \xi \partial _{\xi }\right) ^{l}\left( \frac{\chi _{1}\left( 3\xi \right) \left\{ \eta \right\} ^{\nu }\left| \xi \right| ^{\frac{1}{2}}}{\left| \xi \right| ^{\frac{1}{2}}+\left| \eta \right| ^{\frac{1}{2}}} \theta \left( \xi \eta \right) \right) \right| _{\eta =\mu \left( xt^{-1}\right) }\right) \\&\quad +O\left( \left. \left\{ \xi \right\} ^{-\nu }\left( \xi \partial _{\xi }\right) ^{l}\left( \frac{\chi _{1}\left( 3\xi \right) \left\{ \eta \right\} ^{\nu }\left| \xi \right| ^{\frac{1}{2}}\left( \left| \eta \right| ^{\frac{1}{2}}-\left| \xi \right| ^{\frac{1}{2} }\right) }{\left| \xi \right| +\left| \eta \right| } \theta \left( -\xi \eta \right) \right) \right| _{\eta =\mu \left( xt^{-1}\right) }\right) \le C \end{aligned}$$

and

$$\begin{aligned}&\left| \left\langle \xi \right\rangle ^{\nu }\left\{ \xi \right\} ^{-\nu }\left( \xi \partial _{\xi }\right) ^{l}\mathbf {a}_{2}\left( t,x,\xi \right) \right| \\&\quad =O\left( \left. \left\{ \xi \right\} ^{-\nu }\left( \xi \partial _{\xi }\right) ^{l}\left( \frac{\chi _{1}\left( 3\xi \right) \left\{ \eta \right\} ^{\nu }\left| \xi \right| ^{\frac{1}{2}}\left( 2\left| \eta \right| ^{\frac{1}{2}}+\left| \xi \right| ^{\frac{1}{2}}\right) }{\left( \left| \xi \right| ^{\frac{1}{2} }+\left| \eta \right| ^{\frac{1}{2}}\right) ^{2}}\theta \left( \xi \eta \right) \right) \right| _{\eta =\mu \left( xt^{-1}\right) }\right) \\&\quad +O\left( \left. \left\{ \xi \right\} ^{-\nu }\left( \xi \partial _{\xi }\right) ^{l}\left( \frac{\chi _{1}\left( 3\xi \right) \left\{ \eta \right\} ^{\nu }\left| \xi \right| ^{\frac{1}{2}}\left( \left| \eta \right| ^{\frac{3}{2}}-\frac{1}{2}\left| \xi \right| ^{\frac{3}{2}}-\frac{3}{2}\left| \xi \right| ^{\frac{1}{2}}\left| \eta \right| \right) }{\left( \left| \xi \right| +\left| \eta \right| \right) ^{2}}\theta \left( -\xi \eta \right) \right) \right| _{\eta =\mu \left( xt^{-1}\right) }\right) \le C \end{aligned}$$

for all $x,\xi \in \mathbb {R}$, $t\ge 1,~l=0,1,2$, with some small $\nu >0$. Therefore, applying Lemma 2.4, we find $\left\| \mathbf {a} _{k}\left( t,\xi ,\mathbf {D}\right) \phi \right\| _{_{\mathbf {L}_{\xi }^{2} }}\le C\left\| \phi \right\| _{_{\mathbf {L}^{2}}}$. Thus, we get

$$\begin{aligned} \left\| I_{6}\right\| _{\mathbf {L}^{2}\left( \left| \xi \right| \le 1\right) }&\le \left\| \chi _{1}\left( 3\xi \right) I_{6}\right\| _{\mathbf {L}^{2}}\le C\left\| \mathbf {a}_{1}^{*}\left( t,\xi ,\mathbf {D}\right) \mathcal {F}^{-1}\mathcal {D}_{t} \mathcal {B}M\left\{ \eta \right\} ^{-\nu }\chi _{2}\left( \eta t\right) \partial _{\eta }\phi \left( \eta \right) \right\| _{\mathbf {L}^{2}}\\&\le C\left\| \mathcal {D}_{t}\mathcal {B}M\left\{ \eta \right\} ^{-\nu }\chi _{2}\left( \eta t\right) \partial _{\eta }\phi \left( \eta \right) \right\| _{\mathbf {L}^{2}}\le C\left\| \left| \Lambda ^{\prime \prime }\right| ^{\frac{1}{2}}\left\{ \eta \right\} ^{-\nu }\partial _{\eta }\phi \right\| _{\mathbf {L}^{2}\left( \left| \eta \right| \ge \frac{1}{3t}\right) } \end{aligned}$$

and

$$\begin{aligned} \left\| I_{7}\right\| _{\mathbf {L}^{2}\left( \left| \xi \right| \le 1\right) }&\le \left\| \chi _{1}\left( 3\xi \right) I_{7}\right\| _{\mathbf {L}^{2}}\le \left\| \mathbf {a}_{2}^{*}\left( t,\xi ,\mathbf {D}\right) \mathcal {F}^{-1}\mathcal {D}_{t}\mathcal {B}M\chi _{2}\left( \eta t\right) \frac{\phi \left( \eta \right) }{\eta \left\{ \eta \right\} ^{\nu }}\right\| _{\mathbf {L}^{2}}\\&\le C\left\| \mathcal {D}_{t}\mathcal {B}M\chi _{2}\left( \eta t\right) \frac{\phi \left( \eta \right) }{\eta \left\{ \eta \right\} ^{\nu }}\right\| _{\mathbf {L}^{2}}\le C\left\| \left| \Lambda ^{\prime \prime }\right| ^{\frac{1}{2}}\left\{ \eta \right\} ^{-\nu -1}\phi \right\| _{\mathbf {L} ^{2}\left( \left| \eta \right| \ge t^{-1}\right) }. \end{aligned}$$

Lemma 2.6 is proved. $\square$

3 A priori estimates

Define the norms $\left\| u\right\| _{\mathbf {Z}_{T}}=\sup _{t\in \left[ 0,T\right] }\left\| \left\langle \xi \right\rangle \widehat{\varphi }\right\| _{\mathbf {L}^{\infty }}$ and

$$\begin{aligned} \left\| u\right\| _{\mathbf {Y}_{T}}=\sup _{t\in \left[ 1,T\right] }\left( t^{-\gamma }\left\| \xi \partial _{\xi }\widehat{\varphi }\right\| _{\mathbf {L}^{2}}+t^{-4\gamma }\left\| \partial _{\xi }\widehat{\varphi }\right\| _{\mathbf {L}^{2}}+t^{-\gamma }\left\| \left\langle \xi \right\rangle ^{2}\widehat{\varphi }\right\| _{\mathbf {L}^{2}}\right) , \end{aligned}$$

where $\gamma >0$ is small, $\widehat{\varphi }= \mathcal {FU}\left( -t\right) u\left( t\right)$. Also define $\left\| u\right\| _{\mathbf {X}_{T}}=\left\| u\right\| _{\mathbf {Z}_{T}}+\left\| u\right\| _{\mathbf {Y}_{T}}$.

We first state the local existence of solutions to the Cauchy problem (1.1) (see [4, 23]).

Theorem 3.1

Assume that the initial data $u_{0}\in \mathbf {H}^{2} \cap \mathbf {H}^{0,1}$ and the norm $\varepsilon =\left\| u_{0}\right\| _{\mathbf {H}^{2}\cap \mathbf {H}^{0,1}}$ is sufficiently small. Then, there exists a time $T>1$ such that the Cauchy problem (1.1) has a unique solution $u\in \mathbf {C}\left( \left[ 0,T\right] ;\mathbf {H}^{2} \cap \mathbf {H}^{0,1}\right)$ such that $\left\| u\right\| _{\mathbf {X}_{T}}\le C\varepsilon$.

3.1 $\mathbf {L}^{2}$-norm

The next lemma gives us the estimate of the derivative $\partial _{\xi }\mathcal {FU}\left( -t\right) \left( \left| u\right| ^{2}u\right)$ in the domain $\left| \xi \right| \le 1$.

Lemma 3.1

The estimate $\left\| \partial _{\xi }\mathcal {FU}\left( -t\right) \left( \left| u\right| ^{2}u\right) \right\| _{\mathbf {L}^{2}\left( \left| \xi \right| \le 1\right) }\le Ct^{4\gamma -1}\left\| u\right\| _{\mathbf {X}_{T}}^{3}$ is true for all $t\ge 1$.

Proof

By the factorization techniques, we have $\mathcal {FU}\left( -t\right) \left( \left| u\right| ^{2}u\right) =t^{-1}\mathcal {Q}^{*}\left| v\right| ^{2}v$, $v=\mathcal {Q}\widehat{\varphi }$. Then applying Lemma 2.6 with $\nu =\gamma$, we find

$$\begin{aligned} \left\| \partial _{\xi }\mathcal {FU}\left( -t\right) \left( \left| u\right| ^{2}u\right) \right\| _{\mathbf {L}^{2}\left( \left| \xi \right| \le 1\right) }&\le Ct^{-1}\left\| \left\{ \eta \right\} ^{-\frac{1}{4}}v\right\| _{\mathbf {L}^{2}\left( \left| \eta \right| \ge \frac{1}{3t}\right) }^{2}\left\| \left| \Lambda ^{\prime \prime }\right| ^{\frac{1}{2}}\left\{ \eta \right\} ^{\frac{1}{2}-\gamma }\partial _{\eta }v\right\| _{\mathbf {L}^{2}\left( \left| \eta \right| \ge \frac{1}{3t}\right) }\\&\quad +Ct^{-1}\left\| \left| \Lambda ^{\prime \prime }\right| ^{\frac{1}{2}}\left\{ \eta \right\} ^{-1-\gamma }\left| v\right| ^{2}v\right\| _{\mathbf {L}^{2}\left( \left| \eta \right| \ge \frac{1}{3t}\right) }\\&\quad +Ct^{\gamma -\frac{1}{4}}\left\| v\right\| _{\mathbf {L}^{\infty }\left( \left| \eta \right| \le t^{-1}\right) }^{3}. \end{aligned}$$

Using Lemma 2.5 with $\nu =\gamma$, we get

$$\begin{aligned} \left\| \left| \Lambda ^{\prime \prime }\right| ^{\frac{1}{2} }\left\{ \eta \right\} ^{\frac{1}{2}-\gamma }\partial _{\eta }v\right\| _{\mathbf {L}^{2}\left( \left| \eta \right| \ge \frac{1}{3t}\right) }&\le Ct^{2\gamma }\left\| \xi \partial _{\xi }\widehat{\varphi }\right\| _{\mathbf {L}^{2}}+C\left\| \partial _{\xi }\widehat{\varphi }\right\| _{\mathbf {L}^{2}}\\&\quad +Ct^{\gamma }\left\| \widehat{\varphi }\right\| _{\mathbf {L}^{2} }+Ct^{3\gamma }\left\| \widehat{\varphi }\right\| _{\mathbf {L}^{\infty } }\le Ct^{4\gamma }\left\| u\right\| _{\mathbf {X}_{T}}. \end{aligned}$$

Next, we apply Lemma 2.1, to find

$$\begin{aligned} \left\| \left\{ \eta \right\} ^{-\frac{1}{4}}v\right\| _{\mathbf {L} ^{\infty }\left( \left| \eta \right| \ge \frac{1}{3t}\right) }\le C\left\| u\right\| _{\mathbf {Z}_{T}}+Ct^{4\gamma -\frac{1}{4}}\left\| u\right\| _{\mathbf {Y}_{T}}\le C\left\| u\right\| _{\mathbf {X}_{T}} \end{aligned}$$

and

$$\begin{aligned} \left\| v\right\| _{\mathbf {L}^{\infty }\left( \left| \eta \right| \le t^{-1}\right) }\le Ct^{-\frac{1}{4}}\left\| u\right\| _{\mathbf {Z}_{T}}+Ct^{5\gamma -\frac{1}{2}}\left\| u\right\| _{\mathbf {Y}_{T}}\le Ct^{-\frac{1}{4}}\left\| u\right\| _{\mathbf {X} _{T}}. \end{aligned}$$

Hence,

$$\begin{aligned} \left\| \left| \Lambda ^{\prime \prime }\right| ^{\frac{1}{2} }\left\{ \eta \right\} ^{-1-\gamma }\left| v\right| ^{2}v\right\| _{\mathbf {L}^{2}\left( \left| \eta \right| \ge t^{-1}\right) }&\le C\left\| \left\{ \eta \right\} ^{-\frac{1}{4}}v\right\| _{\mathbf {L} ^{\infty }\left( \frac{1}{3t}\le \left| \eta \right| \le 1\right) }^{3}\left\| \left\{ \eta \right\} ^{-\frac{1}{2}-\gamma }\right\| _{\mathbf {L}^{2}\left( \frac{1}{3t}\le \left| \eta \right| \le 1\right) }\\&\quad +C\left\| v\right\| _{\mathbf {L}^{\infty }\left( \left| \eta \right|>1\right) }^{2}\left\| \left| \Lambda ^{\prime \prime }\right| ^{\frac{1}{2}}v\right\| _{\mathbf {L}^{2}\left( \left| \eta \right| >1\right) }\le Ct^{\gamma }\left\| u\right\| _{\mathbf {X}_{T}}^{3}. \end{aligned}$$

Thus, we obtain $\left\| \partial _{\xi }\mathcal {FU}\left( -t\right) \left( \left| u\right| ^{2}u\right) \right\| _{\mathbf {L} ^{2}\left( \left| \xi \right| \le 1\right) }\le Ct^{4\gamma -1}\left\| u\right\| _{\mathbf {X}_{T}}^{3}$. Lemma 3.1 is proved. $\square$

We prove the a priori estimate in $\mathbf {Y}_{T}$ norm under the condition that the local solution is bounded in $\mathbf {Z}_{T}$.

Lemma 3.2

Let u be the solution stated in Theorem 3.1 and $\left\| u\right\| _{\mathbf {Z}_{T}}\le \varepsilon$. Then, the estimate $\left\| u\right\| _{\mathbf {Y}_{T}}<6\varepsilon$ is true.

Proof

We prove estimate of the lemma by a contradiction. By the continuity, we can find a time $T_{1}<T$ such that $\left\| u\right\| _{\mathbf {Y}_{T_{1}} }=6\varepsilon$. Thus, we have $\left\| u\right\| _{\mathbf {X}_{T_{1}} }\le C\varepsilon$. Applying Lemma 2.1, we get

$$\begin{aligned} \left\| u\right\| _{\mathbf {L}^{\infty }}=t^{-\frac{1}{2}}\left\| \mathcal {Q}\widehat{\varphi }\right\| _{\mathbf {L}^{\infty }}\le t^{-\frac{1}{2}}C\left\| u\right\| _{\mathbf {Z}_{T_{1}}}+Ct^{4\gamma -\frac{3}{4}}\left\| u\right\| _{\mathbf {Y}_{T_{1}}}\le C\varepsilon t^{-\frac{1}{2}}. \end{aligned}$$

for all $t\in \left[ 1,T_{1}\right]$. By the classical energy method, we have

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\left\| u\right\| _{\mathbf {H}^{2}}\le C\left\| u\right\| _{\mathbf {L}^{\infty }}^{2}\left\| u\right\| _{\mathbf {H} ^{2}}\le C\varepsilon ^{2}\left\langle t\right\rangle ^{-1}\left\| u\right\| _{\mathbf {H}^{2}}. \end{aligned}$$

Hence integrating in time, we obtain $\left\| u\right\| _{\mathbf {H} ^{2}}<2\varepsilon \left\langle t\right\rangle ^{\gamma }$. We mention some important identities. The operator $\mathcal {J}=\mathcal {U}\left( t\right) x\mathcal {U}\left( -t\right) =x+it\left| \partial _{x}\right| ^{-\frac{1}{2}}\partial _{x}$ commutes with $\mathcal {L}=i\partial _{t}+\frac{2}{3}\left| \partial _{x}\right| ^{\frac{3}{2}}$, i.e., $\left[ \mathcal {J},\mathcal {L}\right] =0$. Since the symbol $\Lambda \left( \xi \right) =\frac{2}{3}\left| \xi \right| ^{\frac{3}{2}}$ is homogeneous, we can use the operator $\mathcal {P}=x\partial _{x}+\frac{3}{2}t\partial _{t}$, which is related with $\mathcal {J}$ by the identity $\mathcal {P}=\mathcal {J}\partial _{x}-\frac{3}{2}it\mathcal {L}$. So we consider the estimate of $\left\| \mathcal {P}u\right\| _{\mathbf {L}^{2}}$. We have the commutator $\left[ \mathcal {L},\mathcal {P}\right] =\frac{3}{2}\mathcal {L}$, where $\mathcal {L}=i\partial _{t}+\frac{2}{3}\left| \partial _{x}\right| ^{\frac{3}{2}}$. Applying $\mathcal {P}$ to Eq. (1.1) $\mathcal {L}u=\lambda \left| u\right| ^{2}u$ we get $\mathcal {LP}u=\left( \mathcal {P}+\frac{3}{2}\right) \mathcal {L}u=\left( \mathcal {P}+\frac{3}{2}\right) \lambda \left| u\right| ^{2}u$. Hence,

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\left\| \mathcal {P}u\right\| _{\mathbf {L}^{2}}\le C\left\| u\right\| _{\mathbf {L}^{\infty }}^{2}\left( \left\| \mathcal {P}u\right\| _{\mathbf {L}^{2}}+\left\| u\right\| _{\mathbf {L}^{2}}\right) \le C\varepsilon ^{2}t^{-1}\left\| \mathcal {P} u\right\| _{\mathbf {L}^{2}}+C\varepsilon ^{3}t^{\gamma -1} \end{aligned}$$

for $t\in \left[ 1,T_{1}\right]$. Integration with respect to time yields $\left\| \mathcal {P}u\right\| _{\mathbf {L}^{2}}\le \varepsilon +C\varepsilon ^{3}\left\langle t\right\rangle ^{\gamma }$. Then by the identity $\mathcal {P}=\mathcal {J}\partial _{x}-\frac{3}{2}it\mathcal {L}$, we obtain

$$\begin{aligned} \left\| \partial _{x}\mathcal {J}u\right\| _{\mathbf {L}^{2}}\le \left\| \mathcal {P}u\right\| _{\mathbf {L}^{2}}+Ct\left\| u\right\| _{\mathbf {L}^{\infty }}^{2}\left\| u\right\| _{\mathbf {L}^{2} }<2\varepsilon \left\langle t\right\rangle ^{\gamma }. \end{aligned}$$

Finally, let us estimate $\left\| \mathcal {J}u\right\| _{\mathbf {L}^{2} }$. Since

$$\begin{aligned} \left\| \mathcal {J}u\right\| _{\mathbf {L}^{2}}=\left\| \partial _{\xi }\widehat{\varphi }\right\| _{\mathbf {L}^{2}\left( \left| \xi \right| \le 1\right) }+\left\| \partial _{\xi }\widehat{\varphi }\right\| _{\mathbf {L}^{2}\left( \left| \xi \right| \ge 1\right) }\le \left\| \partial _{\xi }\widehat{\varphi }\right\| _{\mathbf {L} ^{2}\left( \left| \xi \right| \le 1\right) }+\left\| \partial _{x}\mathcal {J}u\right\| _{\mathbf {L}^{2}}, \end{aligned}$$

then it is sufficient to estimate the norm $\left\| \partial _{\xi } \widehat{\varphi }\right\| _{\mathbf {L}^{2}\left( \left| \xi \right| \le 1\right) }$. Multiplying equation (1.1) by $\mathcal {FU}\left( -t\right)$, we obtain $i\partial _{t}\widehat{\varphi }=\lambda \mathcal {FU} \left( -t\right) \left( \left| u\right| ^{2}u\right)$ for the function $\widehat{\varphi }= \mathcal {FU}\left( -t\right) u\left( t\right)$. Differentiating we get $i\partial _{t}\partial _{\xi } \widehat{\varphi }=\lambda \partial _{\xi }\mathcal {FU}\left( -t\right) \left( \left| u\right| ^{2}u\right)$. Applying Lemma 3.1, we find $\left\| \partial _{\xi }\mathcal {FU}\left( -t\right) \left( \left| u\right| ^{2}u\right) \right\| _{\mathbf {L}^{2}\left( \left| \xi \right| \le 1\right) }\le C\varepsilon ^{3}t^{4\gamma -1}$ for $t\in \left[ 1,T_{1}\right]$. Then denoting the norm $y=\left\| \partial _{\xi }\widehat{\varphi }\right\| _{\mathbf {L}^{2}\left( \left| \xi \right| \le 1\right) }$ we get $\frac{\mathrm{d}y}{\mathrm{d}t}\le C\varepsilon ^{3}t^{4\gamma -1}$. Integrating in time, we find $y=\left\| \partial _{\xi }\widehat{\varphi }\right\| _{\mathbf {L}^{2}\left( \left| \xi \right| \le 1\right) }<2\varepsilon t^{4\gamma }$ for $t\in \left[ 1,T_{1}\right]$. Thus, we obtain $\left\| u\right\| _{\mathbf {Y} _{T_{1}}}<6\varepsilon$, which yields a desired contradiction. Lemma 3.2 is proved. $\square$

3.2 $\mathbf {L}^{\infty }$-norm

In the next lemma, we calculate the asymptotic representation for $\mathcal {FU}\left( -t\right) \left( \left| u\right| ^{2}u\right)$.

Lemma 3.3

The asymptotic representation

$$\begin{aligned} \mathcal {FU}\left( -t\right) \left( \left| u\right| ^{2}u\right) =\frac{1}{t\Lambda ^{\prime \prime }\left( \xi \right) }\left| \widehat{\varphi }\left( \xi \right) \right| ^{2}\widehat{\varphi }\left( \xi \right) +O\left( \left\langle \xi \right\rangle ^{\frac{1}{8}} t^{12\gamma -\frac{5}{4}}\left\| u\right\| _{\mathbf {X}_{T}}^{3}\right) \end{aligned}$$

is true for all $t\ge 1$, $\xi \in \mathbb {R}$, where $\widehat{\varphi }\left( t\right) =\mathcal {FU}\left( -t\right) u\left( t\right)$.

Proof

By the factorization property, we obtain $\mathcal {FU}\left( -t\right) \left( \left| u\right| ^{2}u\right) =t^{-1}\mathcal {Q}^{*}\left| v\right| ^{2}v$, where $v=\mathcal {Q}\widehat{\varphi }$. By virtue of Lemma 2.2, we get

$$\begin{aligned}&\left\langle \xi \right\rangle ^{-\frac{1}{8}}\mathcal {Q}^{*}\left| v\right| ^{2}v=\left\langle \xi \right\rangle ^{-\frac{1}{8}}A^{*}\left| v\right| ^{2}v+O\left( t^{-\frac{1}{4}}\left\| \left| \eta \right| ^{-\frac{1}{4}}\partial _{\eta }\left| v\right| ^{2}v\right\| _{\mathbf {L}^{2}\left( \left| \eta \right| \ge \frac{1}{3t}\right) }\right) \\&+O\left( t^{-\frac{1}{4}}\left\| \left| \eta \right| ^{-\frac{5}{4}}\left| v\right| ^{2}v\right\| _{\mathbf {L}^{2}\left( \left| \eta \right| \ge \frac{1}{3t}\right) }\right) +O\left( \left\| v\right\| _{\mathbf {L}^{\infty }\left( \left| \eta \right| \le t^{-1}\right) }^{3}\right) . \end{aligned}$$

As in the proof of Lemma 3.1, we get

$$\begin{aligned} \left\| \left| \eta \right| ^{-\frac{1}{4}}\partial _{\eta }\left| v\right| ^{2}v\right\| _{\mathbf {L}^{2}\left( \left| \eta \right| \ge \frac{1}{3t}\right) }\le C\left\| \left\{ \eta \right\} ^{-\frac{1}{4}}v\right\| _{\mathbf {L}^{2}\left( \left| \eta \right| \ge \frac{1}{3t}\right) }^{2}\left\| \left| \Lambda ^{\prime \prime }\right| ^{\frac{1}{2}}\left\{ \eta \right\} ^{\frac{1}{2}}\partial _{\eta }v\right\| _{\mathbf {L}^{2}\left( \left| \eta \right| \ge \frac{1}{3t}\right) }\le Ct^{4\gamma }\left\| u\right\| _{\mathbf {X}_{T}}^{3}, \end{aligned}$$

$$\begin{aligned} \left\| \left| \eta \right| ^{-\frac{5}{4}}\left| v\right| ^{2}v\right\| _{\mathbf {L}^{2}\left( \left| \eta \right| \ge \frac{1}{3t}\right) }\le Ct^{\gamma }\left\| u\right\| _{\mathbf {X} _{T}}^{3} \end{aligned}$$

and $\left\| v\right\| _{\mathbf {L}^{\infty }\left( \left| \eta \right| \le t^{-1}\right) }^{3}\le Ct^{-\frac{3}{4}}\left\| u\right\| _{\mathbf {X}_{T}}$. Thus, we get in view of the asymptotics of the kernel $A^{*}$

$$\begin{aligned} \mathcal {Q}^{*}\left| v\right| ^{2}v=\sqrt{i\Lambda ^{\prime \prime }\left( \xi \right) }\left| v\right| ^{2}v+O\left( t^{4\gamma -\frac{1}{4}}\left\langle \xi \right\rangle ^{\frac{1}{8}}\left\| u\right\| _{\mathbf {X}_{T}}^{3}\right) . \end{aligned}$$

Then, we apply Lemma 2.1 to represent the first summand on the right-hand side of the above formula in the form

$$\begin{aligned} \sqrt{i\Lambda ^{\prime \prime }\left( \xi \right) }\left| v\right| ^{2}v=\frac{1}{\Lambda ^{\prime \prime }\left( \xi \right) }\left| \widehat{\varphi }\left( \xi \right) \right| ^{2}\widehat{\varphi }\left( \xi \right) +O\left( t^{12\gamma -\frac{1}{4}}\left\| u\right\| _{\mathbf {X}_{T}}^{3}\right) . \end{aligned}$$

Hence, we get

$$\begin{aligned} \mathcal {Q}^{*}\left| v\right| ^{2}v=\frac{1}{\Lambda ^{\prime \prime }\left( \xi \right) }\left| \widehat{\varphi }\left( \xi \right) \right| ^{2}\widehat{\varphi }\left( \xi \right) +O\left( \left\langle \xi \right\rangle ^{\frac{1}{8}}t^{12\gamma -\frac{1}{4}}\left\| u\right\| _{\mathbf {X}_{T}}^{3}\right) . \end{aligned}$$

Lemma 3.3 is proved. $\square$

We next prove a priori estimate of the local solutions in $\mathbf {Z}_{T}$ norm under the boundedness condition in $\mathbf {Y}_{T}$.

Lemma 3.4

Let u be the solution stated in Theorem 3.1 and $\left\| u\right\| _{\mathbf {Y}_{T}}\le \varepsilon$. Then, the estimate $\left\| u\right\| _{\mathbf {Z}_{T}}<2\varepsilon$ is true.

Proof

In the domain $\left| \xi \right| \ge \left\langle t\right\rangle ^{\nu }$, we get by the Sobolev embedding inequality

$$\begin{aligned} \left\| \left\langle \xi \right\rangle \widehat{\varphi }\right\| _{\mathbf {L}^{\infty }\left( \left| \xi \right| \ge \left\langle t\right\rangle ^{\nu }\right) }&\le \sqrt{2}\left\| \left\langle \xi \right\rangle \widehat{\varphi }\right\| _{\mathbf {L}^{2}\left( \left| \xi \right| \ge \left\langle t\right\rangle ^{\nu }\right) }^{\frac{1}{2}}\left\| \partial _{\xi }\left\langle \xi \right\rangle \widehat{\varphi }\right\| _{\mathbf {L}^{2}\left( \left| \xi \right| \ge \left\langle t\right\rangle ^{\nu }\right) }^{\frac{1}{2}}\\&\le \sqrt{2}\left\langle t\right\rangle ^{-\frac{\nu }{2}}\left\| \left\langle \xi \right\rangle ^{2}\widehat{\varphi }\right\| _{\mathbf {L} ^{2}\left( \left| \xi \right| \ge \left\langle t\right\rangle ^{\nu }\right) }^{\frac{1}{2}}\left\| \partial _{\xi }\left\langle \xi \right\rangle \widehat{\varphi }\right\| _{\mathbf {L}^{2}\left( \left| \xi \right| \ge \left\langle t\right\rangle ^{\nu }\right) }^{\frac{1}{2} }<2\varepsilon \left\langle t\right\rangle ^{-\frac{\nu }{2}+\frac{3}{2}\gamma }<2\varepsilon \end{aligned}$$

if $\frac{\nu }{2}>\frac{3}{2}\gamma$. Therefore, we need to estimate the function $\left\langle \xi \right\rangle \widehat{\varphi }\left( t,\xi \right)$ in the domain $\left| \xi \right| \le \left\langle t\right\rangle ^{\nu }$. Applying the operator $\mathcal {FU}\left( -t\right)$ to Eq. (1.1) $\mathcal {L}u=\lambda \left| u\right| ^{2}u$, we get $i\widehat{\varphi }_{t}\left( t,\xi \right) =\lambda \mathcal {FU}\left( -t\right) \left( \left| u\right| ^{2}u\right)$. By virtue of Lemma 3.3, we obtain

$$\begin{aligned} i\left\langle \xi \right\rangle \widehat{\varphi }_{t}\left( t,\xi \right) =\frac{\lambda }{\Lambda ^{\prime \prime }\left( \xi \right) }\left| \widehat{\varphi }\left( \xi \right) \right| ^{2}\left\langle \xi \right\rangle \widehat{\varphi }\left( \xi \right) +O\left( \left\langle \xi \right\rangle ^{\frac{5}{4}}t^{12\gamma -\frac{5}{4}}\left\| u\right\| _{\mathbf {X}_{T}}^{3}\right) . \end{aligned}$$

Multiplying by $\left\langle \xi \right\rangle \overline{\widehat{\varphi } _{t}\left( t,\xi \right) }$ , and taking the imaginary part, we get $\frac{\mathrm{d}}{\mathrm{d}t}\left| \left\langle \xi \right\rangle \widehat{\varphi }\left( t,\xi \right) \right| ^{2}\le C\varepsilon ^{4}t^{\frac{5}{4}\nu +12\gamma -\frac{5}{4}}$ in the domain $\left| \xi \right| \le \left\langle t\right\rangle ^{\nu }$. Define $t_{1}$ such that $\left\langle t_{1}\right\rangle ^{\nu }=\left| \xi \right|$, then integrating in time from $t_{1}$ to t, we obtain

$$\begin{aligned} \left| \left\langle \xi \right\rangle \widehat{\varphi }\left( t,\xi \right) \right| ^{2}\le \left| \left\langle \xi \right\rangle \widehat{\varphi }\left( t_{1},\xi \right) \right| ^{2}+C\varepsilon ^{4}\int _{t_{1}}^{t}\tau ^{\frac{5}{4}\nu +12\gamma -\frac{5}{4}}\mathrm{d}\tau <2\varepsilon ^{2}. \end{aligned}$$

Lemma 3.4 is proved. $\square$

4 Proof of Theorem 1.1

By Lemma 3.2, we see that a priori estimate of $\left\| u\right\| _{\mathbf {Z}_{T}}$ implies a priori estimate of $\left\| u\right\| _{\mathbf {Y}_{T}}$. On the other hand, by Lemma 3.4, a priori estimate of $\left\| u\right\| _{\mathbf {Y}_{T}}$ yields a priori estimate of $\left\| u\right\| _{\mathbf {Z}_{T}}$. Therefore, global existence of solutions of the Cauchy problem (1.1) satisfying estimates $\left\| u\right\| _{\mathbf {X}_{\infty }}\le C\varepsilon$ follow by a standard continuation argument via the local existence Theorem 3.1. Thus, we have the global in time existence of solutions to the Cauchy problem (1.1).

Now, we turn to the proof of the asymptotic formula (1.3) for the solutions u of the Cauchy problem (1.1). By the factorization formula $u\left( t\right) =\mathcal {D}_{t}\mathcal {B}M\mathcal {Q}\widehat{\varphi }$ and Lemma 2.1, we find $u\left( t\right) =\mathcal {D} _{t}\mathcal {B}M\frac{1}{\sqrt{i\Lambda ^{\prime \prime }}}\widehat{\varphi }+C\varepsilon t^{-\frac{1}{4}+3\gamma }$. As in the proof of Lemma 3.4, we get $\left\| \left\langle \xi \right\rangle \widehat{\varphi }\right\| _{\mathbf {L}^{\infty }\left( \left| \xi \right| \ge \left\langle t\right\rangle ^{\nu }\right) }\le C\varepsilon \left\langle t\right\rangle ^{-\frac{\nu }{2}+\frac{3}{2}\gamma }$. So we need to compute the asymptotics of the function $\frac{1}{\sqrt{i\Lambda ^{\prime \prime }}}\widehat{\varphi }$ in the domain $\left| \xi \right| \le \left\langle t\right\rangle ^{\nu }$. As in the proof of Lemma 3.4, we get

$$\begin{aligned} \frac{1}{\sqrt{i\Lambda ^{\prime \prime }}}\partial _{t}\widehat{\varphi }\left( t,\xi \right) =-\frac{i\lambda }{\Lambda ^{\prime \prime }\left( \xi \right) }\left| \widehat{\varphi }\left( \xi \right) \right| ^{2}\frac{1}{\sqrt{i\Lambda ^{\prime \prime }}}\widehat{\varphi }\left( \xi \right) +O\left( \varepsilon ^{3}t^{\frac{1}{2}\nu +12\gamma -\frac{5}{4}}\right) . \end{aligned}$$

Then, we change the dependent variable $\frac{1}{\sqrt{i\Lambda ^{\prime \prime } }}\widehat{\varphi }\left( t,\xi \right) =y\left( t,\xi \right) \Psi \left( t,\xi \right)$ with $\Psi \left( t,\xi \right) =\exp \left( -\frac{i\lambda }{\Lambda ^{\prime \prime }\left( \xi \right) }\int _{1}^{t}\left| \widehat{\varphi }\left( \tau ,\xi \right) \right| ^{2}\frac{\mathrm{d}\tau }{\tau }\right)$, to get $\partial _{t}y\left( t,\xi \right) =O\left( \varepsilon ^{3}t^{\frac{1}{2}\nu +12\gamma -\frac{5}{4}}\right)$. Integration in time yields $\left\| y\left( t\right) -y\left( s\right) \right\| _{\mathbf {L}^{\infty }}\le C\int _{s}^{t}\left( \varepsilon ^{3}\tau ^{\frac{1}{2}\nu +12\gamma -\frac{5}{4}}\right) \mathrm{d}\tau \le C\varepsilon s^{-\delta _{1} }$ for all $t>s>0$, with $\delta _{1}=\frac{1}{4}-\frac{1}{2}\nu -12\gamma >0$. Therefore, there exists a unique final state $y_{+}\in \mathbf {L}^{\infty }$ such that $\left\| y\left( t\right) -y_{+}\right\| _{\mathbf {L}^{\infty } }\le C\varepsilon t^{-\delta _{1}}$ for all $t>0$. Since $\left| \frac{1}{\sqrt{i\Lambda ^{\prime \prime }}}\widehat{\varphi }\left( \tau ,\xi \right) \right| ^{2}=\left| y\left( t,\xi \right) \right| ^{2}$, we have $\frac{1}{\Lambda ^{\prime \prime }\left( \xi \right) }\int _{1}^{t}\left| \widehat{\varphi }\left( \tau ,\xi \right) \right| ^{2}\frac{\mathrm{d}\tau }{\tau }=\int _{1}^{t}\left| y\left( \tau ,\xi \right) \right| ^{2}\frac{\mathrm{d}\tau }{\tau }$. Denote $\Phi \left( t\right) =\int _{1}^{t}\left| y\left( \tau \right) \right| ^{2}\frac{\mathrm{d}\tau }{\tau }-\left| y_{+}\right| ^{2}\log t$. We study the asymptotics in time of the remainder term $\Phi \left( t\right)$. We have

$$\begin{aligned} \Phi \left( t\right) -\Phi \left( s\right) =\int _{s}^{t}\left( \left| y\left( \tau \right) \right| ^{2}-\left| y\left( t\right) \right| ^{2}\right) \frac{\mathrm{d}\tau }{\tau }+\left( \left| y\left( t\right) \right| ^{2}-\left| y_{+}\right| ^{2}\right) \log \frac{t}{s} \end{aligned}$$

and $\left\| \Phi \left( t\right) -\Phi \left( s\right) \right\| _{\mathbf {L}^{\infty }}\le C\varepsilon ^{2}s^{-\delta _{1}}$ for all $t>s>0$. Hence, there exists a unique real-valued function $\Phi _{+}$ such that $\Phi _{+}\in \mathbf {L}^{\infty }$ and $\left\| \Phi \left( t\right) -\Phi _{+}\right\| _{\mathbf {L}^{\infty }}\le C\varepsilon ^{2}t^{-\delta _{1}}$. Therefore, we obtain

$$\begin{aligned} \frac{1}{\Lambda ^{\prime \prime }\left( \xi \right) }\int _{1}^{t}\left| \widehat{\varphi }\left( \tau ,\xi \right) \right| ^{2}\frac{\mathrm{d}\tau }{\tau }=\int _{1}^{t}\left| y\left( \tau ,\xi \right) \right| ^{2}\frac{\mathrm{d}\tau }{\tau }=\Phi _{+}+\left| y_{+}\right| ^{2}\log t+O\left( \varepsilon ^{2}t^{-\delta _{1}}\right) \end{aligned}$$

for all $t>0$. Then, we obtain $\left\| \Psi \left( t,\xi \right) -\exp \left( -i\lambda \left| y_{+}\right| ^{2}\log t-i\lambda \Phi _{+}+O\left( \varepsilon ^{2}t^{-\delta _{1}}\right) \right) \right\| _{\mathbf {L}^{\infty }}\le C\varepsilon ^{2}t^{-\delta _{1}}$ for all $t>0$. Thus, we get the large-time asymptotics

$$\begin{aligned} \frac{1}{\sqrt{i\Lambda ^{\prime \prime }}}\widehat{\varphi }\left( t,\xi \right)&=y\left( t,\xi \right) \Psi \left( t,\xi \right) =y_{+} \Psi \left( t,\xi \right) +O\left( \varepsilon t^{-\delta _{1}}\right) \\&=W_{+}\exp \left( -i\lambda \left| W_{+}\right| ^{2}\log t\right) +O\left( \varepsilon t^{-\delta _{1}}\right) , \end{aligned}$$

where $W_{+}=y_{+}\exp \left( -i\lambda \Phi _{+}\right)$. Note that $W_{+} \in \mathbf {L}^{\infty }$. Using the factorization of $\mathcal {U}\left( t\right)$, we have

$$\begin{aligned} u\left( t\right) =\mathcal {D}_{t}\mathcal {B}M\frac{1}{\sqrt{i\Lambda ^{\prime \prime }}}\widehat{\varphi }+C\varepsilon t^{-\frac{1}{4}+3\gamma }=\mathcal {D}_{t}\mathcal {B}MW_{+}\exp \left( -i\lambda \left| W_{+}\right| ^{2}\log t\right) +O\left( \varepsilon t^{-\delta _{1} }\right) . \end{aligned}$$

This completes the proof of the asymptotics (1.3). Theorem 1.1 is proved.